Antisymmetric Orbit Functions
In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space En are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of s...
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Klimyk, A. Patera, J. 2019-02-16T08:08:46Z 2019-02-16T08:08:46Z 2007 Antisymmetric Orbit Functions / A. Klimyk, J. Patera // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 39 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33-02; 33E99; 42B99; 42C15; 58C40 https://nasplib.isofts.kiev.ua/handle/123456789/147784 In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space En are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group G of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space En. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in En, vanishing on the boundary of the fundamental domain F. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group G. They also determine a transform on a finite set of points of F (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given. The first author (AK) acknowledges CRM of University of Montreal for hospitality when this paper was under preparation. His research was partially supported by Grant 10.01/015 of the State Foundation of Fundamental Research of Ukraine. We are grateful for partial support for this work to the National Research Council of Canada, MITACS, the MIND Institute of Costa Mesa, California, and Lockheed Martin, Canada. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Antisymmetric Orbit Functions Article published earlier |
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In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space En are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group G of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space En. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in En, vanishing on the boundary of the fundamental domain F. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group G. They also determine a transform on a finite set of points of F (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.
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Antisymmetric Orbit Functions / A. Klimyk, J. Patera // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 39 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 023, 83 pages
Antisymmetric Orbit Functions
Anatoliy KLIMYK † and Jiri PATERA ‡
† Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv 03143, Ukraine
E-mail: aklimyk@bitp.kiev.ua
‡ Centre de Recherches Mathématiques, Université de Montréal,
C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada
E-mail: patera@crm.umontreal.ca
Received December 25, 2006; Published online February 12, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/023/
Abstract. In the paper, properties of antisymmetric orbit functions are reviewed and
further developed. Antisymmetric orbit functions on the Euclidean space En are antisym-
metrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corre-
sponding to a Coxeter–Dynkin diagram. Properties of such functions are described. These
functions are closely related to irreducible characters of a compact semisimple Lie group G
of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies
of the fundamental domain F of the affine Weyl group (determined by the initial Weyl
group) in the entire Euclidean space En. Antisymmetric orbit functions are solutions of
the corresponding Laplace equation in En, vanishing on the boundary of the fundamental
domain F . Antisymmetric orbit functions determine a so-called antisymmetrized Fourier
transform which is closely related to expansions of central functions in characters of irre-
ducible representations of the group G. They also determine a transform on a finite set of
points of F (the discrete antisymmetric orbit function transform). Symmetric and antisym-
metric multivariate exponential, sine and cosine discrete transforms are given.
Key words: antisymmetric orbit functions; signed orbits; products of orbits; orbit function
transform; finite orbit function transform; finite Fourier transforms; finite cosine transforms;
finite sine transforms; symmetric functions
2000 Mathematics Subject Classification: 33-02; 33E99; 42B99; 42C15; 58C40
1 Introduction
In [1] and [2] we considered properties and applications of symmetric orbit functions (they are
called there orbit functions; without the word “symmetric”). They are closely related to finite
groups W of geometric symmetries generated by reflection transformations ri (that is, such
that r2i = 1), i = 1, 2, . . . , n, of the n-dimensional Euclidean space En with respect to (n−1)-
dimensional subspaces containing the origin. In order to obtain a symmetric orbit function we
take a point λ ∈ En and act upon λ by all elements of the group W . If O(λ) is the W -orbit of
the point λ, that is the set of all different points of the form wλ, w ∈ W , then the symmetric
orbit function, determined by λ, coincides with
φλ(x) =
∑
µ∈O(λ)
e2πi〈µ,x〉,
where 〈µ, x〉 is the scalar product on En. These functions are invariant with respect to the action
by elements of the group W : φλ(wx) = φλ(x), w ∈ W . If λ is an integral point of En, then
φλ(x) is invariant with respect to the affine Weyl group W aff corresponding to W . Since in the
simplest case symmetric orbit functions coincide with the cosine function, sometimes they are
called C-functions.
mailto:aklimyk@bitp.kiev.ua
mailto:patera@crm.umontreal.ca
http://www.emis.de/journals/SIGMA/2007/023/
2 A. Klimyk and J. Patera
Symmetric orbit functions are multivariate functions having many beautiful and useful pro-
perties and applicable both in mathematics and engineering. For this reason, they can be treated
as special functions [2].
Symmetricity is the main property of symmetric orbit functions, considered in [1] and [2],
which make them useful in applications. Being a modification of monomial symmetric functions,
they are directly related to the theory of symmetric (Laurent) polynomials [3, 4, 5, 6] (see
Section 11 in [1]).
Symmetric orbit functions φλ(x) for integral λ are closely related to the representation theory
of compact groups G. In particular, they were effectively used for different calculations in
representation theory [7, 8, 9, 10, 11]. They are constituents of traces (characters) of irreducible
unitary representations of G. Although characters contain all (or almost all) information about
the corresponding irreducible representations, they are seldom used as special functions. The
reason is that a construction of characters is rather complicated, whereas orbit functions have
much more simple structure.
The symmetric orbit function φλ(x) is a symmetrized (by means of the group W ) exponential
function e2πi〈λ,x〉 on En. For each transformation group W , the symmetric orbit functions
form a complete orthogonal basis in the space of symmetric (with respect to W ) polynomials
in e2πixj , j = 1, 2, . . . , n, or in the Hilbert space obtained by closing this space with respect to an
appropriate scalar product. Orbit functions φλ(x), when λ runs over integral weights, determine
so-called (symmetric) orbit function transform, which is a symmerization of the usual Fourier
series expansion on En. If λ runs over the space En, then φλ(x) determines a symmetric orbit
function transform, which is a symmetrization of the usual continuous Fourier expansion in En
(that is, of the Fourier integral).
In the same way as the Fourier transform leads to discrete Fourier transform, the symmetric
orbit function transform leads to a discrete analogue of this transform (which is a generalization
of the discrete cosine transform [12]). This discrete transform is useful in many things related
to discretization (see [13, 14, 15, 16, 17, 18, 19, 20]). Construction of the discrete orbit function
transform is fulfilled by means of the results of paper [21].
In this paper we consider antisymmetric orbit functions (since in the simplest case they
coincide with the sine function, sometimes they are called S-functions). They are given by
ϕλ(x) =
∑
w∈W
(detw)e2πi〈wλ,x〉, x ∈ En,
where λ is a strictly dominant weight and detw is a determinant of the transformation w (it is
equal to 1 or −1, depending on either w is a product of even or odd number of reflections). The
orbit functions ϕλ have many properties that the symmetric orbit functions φλ do. But anti-
symmetricity leads to some new properties which are useful for applications [22]. For integral λ,
antisymmetric orbit functions are closely related to characters of irreducible representations of
the corresponding compact Lie group G. Namely, the character χλ of the irreducible represen-
tation Tλ, λ ∈ P+, coincides with ϕλ+ρ/ϕρ, where ρ is a half of a sum of positive roots related
to the Weyl group W .
Symmetric orbit functions are a generalization of the cosine function, whereas antisymmetric
orbit functions are a generalization of the sine function. A generalization of the exponential
functions are called E-orbit functions. A detailed description of these functions for rank two see
in [23].
Our goal in this paper is to bring together in full generality the diverse facts about antisym-
metric orbit functions, many of which are not found in the literature, although they often are
straightforward consequences of known facts. In general, for a given transformation group W
of the Euclidean space En, most of the properties of antisymmetric orbit functions, which are
Antisymmetric Orbit Functions 3
described in this paper, are implications of properties either of the orbits of the group W or of
the usual exponential function on En.
For dominant elements λ, antisymmetric orbit functions ϕλ(x) are antisymmetric with re-
spect to elements of the corresponding Weyl group, that is, ϕλ(wx) = (detw)ϕλ(x) for any
w ∈ W . For this reason, antisymmetric orbit functions are defined only for strictly dominant
elements λ (a dominant element λ is strictly dominant if wλ = λ means that w = 1). If λ is
integral strictly dominant element, then the corresponding antisymmetric orbit function ϕλ(x)
is antisymmetric also with respect to elements of the affine Weyl group W aff , corresponding
to the Weyl group W . Antisymmetricity is a main property of antisymmetric orbit functions.
Because of antisymmetricity, it is enough to determine ϕλ(x) only on a fundamental domain of
the affine Weyl group W aff (if λ is integral).
In the case when the group W is a direct product of its subgroups, say W = W1 ×W2, the
fundamental domain is the Cartesian product of fundamental domains for W1 and W2. Similarly,
antisymmetric orbit functions of W are products of antisymmetric orbit functions of W1 and W2.
Hence it suffices to carry out our considerations for groups W which cannot be represented as
a product of its subgroups (that is, for such W for which a corresponding Coxeter–Dynkin
diagram is connected).
In the article many examples of dimensions 2 and 3 are shown because they are likely to be
used more often.
We shall need a general information on Weyl groups, affine Weyl groups, root systems and
their properties. We have given this information in [1]. In order to make this paper self-contained
we repeat a part of that information in Section 2.
In Section 3 we describe signed Weyl group orbits. They differ from the Weyl group or-
bits by a sign (equal to +1 or −1) assigned to each point of an orbit. To each signed orbit
there corresponds an antisymmetric orbit function if a dominant element of the orbit is strictly
dominant.
Section 4 is devoted to description of antisymmetric orbit functions. Antisymmetric orbit
functions, corresponding to Coxeter–Dynkin diagrams, containing only two nodes, are given in
an explicit form. In this section we also give explicit formulas for antisymmetric orbit functions,
corresponding to Coxeter–Dynkin diagrams of An, Bn, Cn and Dn, in the corresponding ortho-
gonal coordinate systems. In Section 5 properties of antisymmetric orbit functions are described.
If λ is integral, then a main property of the antisymmetric orbit function ϕλ is an invariance
with respect to the affine Weyl group W aff . We also give here the symmetric and antisymmetric
orbit functions φρ(x) and ϕρ(x), corresponding to the half-sum ρ of positive roots, in a form of
products of the cosine and sine functions of certain angles depending on x. Specific properties
of antisymmetric orbit functions of the Coxeter–Dynkin diagram An are given in Section 6.
In Section 7 we consider expansions of products of symmetric (antisymmetric) orbit functions
into a sum of symmetric or antisymmetric orbit functions. These expansions are closely related
to properties of (signed) W -orbits. Many examples for expansions in the case of Coxeter–Dynkin
diagrams A2 and C2 are considered. Section 8 is devoted to expansion of antisymmetric W -orbit
functions into a sum of antisymmetric W ′-orbit functions, where W ′ is a subgroup of the Weyl
group W . Many particular cases are studied in detail.
Connection between antisymmetric orbit functions ϕλ(x) with integral λ and characters of
finite dimensional irreducible representations of the corresponding simple compact Lie groups
is studied in Section 9. In particular, the well-known Weyl formula for characters of such
representations contains antisymmetric orbit functions.
In Section 10 we expose antisymmetric orbit function transforms. There are two types of
such transforms. The first one is an analogue of the expansion into Fourier series and the
second one is an analogue of the Fourier integral transform. In Section 11 a description of an
antisymmetric generalization of the multi-dimensional finite Fourier transform is given. This
4 A. Klimyk and J. Patera
analogue is connected with grids on the corresponding fundamental domains for the affine Weyl
groups W aff . Symmetric and antisymmetric multivariate exponential discrete transforms, as
well as symmetric and antisymmetric multivariate sine and cosine discrete transforms are given
in this section.
In Section 12 we show that antisymmetric orbit functions are solutions of the Laplace equation
on the corresponding n-dimensional simplex vanishing on a boundary of the simplex. It is shown
that antisymmetric orbit functions are eigenfunctions of other differential operators.
Section 13 is devoted to exposition of symmetric and antisymmetric functions, which are
symmetric and antisymmetric analogues of special functions of mathematical physics or ortho-
gonal polynomials. In particular, we find eigenfunctions of antisymmetric and symmetric orbit
function transforms. These eigenfunctions are connected with classical Hermite polynomials.
2 Weyl groups and affine Weyl groups
2.1 Coxeter–Dynkin diagrams and Cartan matrices
The sets of symmetric or antisymmetric orbit functions on the n-dimensional Euclidean space En
are determined by finite transformation groups W , generated by reflections ri, i = 1, 2, . . . , n
(a characteristic property of reflections is the equality r2i = 1); the theory of such groups see,
for example, in [24] and [25]. We are interested in those groups W which are Weyl groups
of semisimple Lie groups (semisimple Lie algebras). It is well-known that such Weyl groups
together with the corresponding systems of reflections ri, i = 1, 2, . . . , n, are determined by
Coxeter–Dynkin diagrams. There are 4 series and 5 separate simple Lie algebras, which uniquely
determine their Weyl groups W . They are denoted as
An (n ≥ 1), Bn (n ≥ 3), Cn (n ≥ 2), Dn (n ≥ 4), E6, E7, E8, F4, G2.
To these Lie algebras there correspond connected Coxeter–Dynkin diagrams. To semisimple
Lie algebras (they are direct sums of simple Lie subalgebras) there correspond Coxeter–Dynkin
diagrams, which consist of connected parts, corresponding to simple Lie subalgebras; these
parts are not connected with each other (a description of the correspondence between simple
Lie algebras and Coxeter–Dynkin diagrams see, for example, in [26]). Thus, we describe only
Coxeter–Dynkin diagrams, corresponding to simple Lie algebras. They are of the form
An g1 g2 g3 gn· · ·
Bn g1 g2 gn−1 wn· · ·
Cn w1 w2 wn−1 gn· · ·
Dn
g1 g2 gn−3 g
n−2
gn−1gn
· · ·
E6
g1 g2 g
3
g4 g5g6
E7
g1 g2 g
3
g4 g5 g6g7
E8
g1 g2 g3 g4 g
5
g6 g7g8
F4
g1 g2 w3 w4 G2
g1 w2
A diagram determines a certain non-orthogonal basis {α1, α2, . . . , αn} in the Euclidean spa-
ce En. Each node is associated with a basis vector αk, called a simple root. A direct link
between two nodes indicates that the corresponding basis vectors are not orthogonal. Conversely,
Antisymmetric Orbit Functions 5
an absence of a direct link between nodes implies orthogonality of the corresponding vectors.
Single, double, and triple links indicate that the relative angles between the two simple roots
are 2π/3, 3π/4, 5π/6, respectively. There can be only two cases: all simple roots are of the same
length or there are only two different lengths of simple roots. In the first case all simple roots
are denoted by white nodes. In the case of two lengths, shorter roots are denoted by black nodes
and longer ones by white nodes. Lengths of roots are determined uniquely up to a common
constant. For the cases Bn, Cn, and F4, the squared longer root length is double the squared
shorter root length. For G2, the squared longer root length is triple the squared shorter root
length.
If two nodes are connected by a single line, then the angle between the corresponding simple
roots is 2π/3. If nodes are connected by a double line, then the angle is 3π/4. A triple line
means that the angle is 5π/6. Simple roots of the same length are orthogonal to each other or
an angle between them is 2π/3.
To each Coxeter–Dynkin diagram there corresponds a Cartan matrix M , consisting of the
entries
Mjk =
2〈αj , αk〉
〈αk, αk〉
, j, k ∈ {1, 2, . . . , n}, (2.1)
where 〈x, y〉 denotes the scalar product of x, y ∈ En. Cartan matrices of simple Lie algebras are
given in many places (see, for example, [27]). We recall them here for ranks 2 and 3 because of
their usage later on:
A2 :
(
2 −1
−1 2
)
, C2 :
(
2 −1
−2 2
)
, G2 :
(
2 −3
−1 2
)
,
A3 :
2 −1 0
−1 2 −1
0 −1 2
, B3 :
2 −1 0
−1 2 −2
0 −1 2
, C3 :
2 −1 0
−1 2 −1
0 −2 2
.
Lengths of the basis vectors αi are fixed by the corresponding Coxeter–Dynkin diagram up
to a constant. We adopt the standard choice in the Lie theory, namely
〈α, α〉 = 2
for all simple roots of An, Dn, E6, E7, E8 and for the longer simple roots of Bn, Cn, F4, G2.
2.2 Weyl group
A Coxeter–Dynkin diagram determines uniquely the corresponding transformation group of En,
generated by reflections ri, i = 1, 2, . . . , n. These reflections correspond to simple roots αi,
i = 1, 2, . . . , n. Namely, the transformation ri corresponds to the simple root αi and is the
reflection with respect to (n − 1)-dimensional linear subspace (hyperplane) of En (containing
the origin), orthogonal to αi. It is well-known that such reflections are given by the formula
rix = x− 2〈x, αi〉
〈αi, αi〉
αi, i = 1, 2, . . . , n, x ∈ En. (2.2)
Each reflection ri can be thought as attached to the i-th node of the corresponding diagram.
A finite group W , generated by the reflections ri, i = 1, 2, . . . , n, is called a Weyl group,
corresponding to a given Coxeter–Dynkin diagram. If a Weyl groupW corresponds to a Coxeter–
Dynkin diagram of a simple Lie algebra L, then this Weyl group is often denoted by W (L).
Properties of Weyl groups are well known (see [24] and [25]). The orders (numbers of elements)
of Weyl groups are given by the formulas
|W (An)| = (n+ 1)!, |W (Bn)| = |W (Cn)| = 2nn!, |W (Dn)| = 2n−1n!,
6 A. Klimyk and J. Patera
|W (E6)| = 51 840, |W (E7)| = 2903 040, |W (E8)| = 696 729 600, (2.3)
|W (F4)| = 1152, |W (G2)| = 12.
In particular,
|W (A2)| = 6, |W (C2)| = 8, |W (A3)| = 24, |W (C3)| = 48.
2.3 Roots and weights
A Coxeter–Dynkin diagram determines a system of simple roots in the Euclidean space En.
Acting by elements of the Weyl group W upon simple roots we obtain a finite system of vectors,
which is invariant with respect to W . A set of all these vectors is called a system of roots
associated with a given Coxeter–Dynkin diagram. It is denoted by R. As we see, a system of
roots R is calculated from simple roots by a straightforward algorithm.
It is proved (see, for example, [26]) that roots of R are linear combinations of simple roots
with integral coefficients. Moreover, there exist no roots, which are linear combinations of αi,
i = 1, 2, . . . , n, both with positive and negative coefficients. Therefore, the set of roots R can
be represented as a union R = R+ ∪ R−, where R+ (respectively R−) is the set of roots which
are linear combination of simple roots with positive (negative) coefficients. The set R+ (the
set R−) is called a set of positive (negative) roots.
As mentioned above, a set R of roots is invariant under the action of elements of the Weyl
group W (R). However, wR+ 6= R+ if w is not a trivial element of W . The following proposition
holds:
Proposition 1. A reflection ri ∈W , corresponding to a simple root αi, maps αi into −αi and
reflects the set R+\{αi} of all other roots of R+ onto itself.
Let Xα be the (n − 1)-dimensional linear subspace (hyperplane) of En which contains the
origin and is orthogonal to the root α. Clearly, Xα = X−α. The set of reflections with respect
to Xα, α ∈ R+, coincides with the set of all reflections of the corresponding Weyl group W .
The hyperplane Xα consists of all points x ∈ En such that 〈x, α〉 = 0.
The subspaces Xα, α ∈ R+, split the Euclidean space En into connected parts which are
called Weyl chambers. A number of Weyl chambers coincides with the number of elements of
the Weyl group W . Elements of the Weyl group permute Weyl chambers. A part of a Weyl
chamber, which belongs to some hyperplane Xα is called a wall of this Weyl chamber. If for
some element x of a Weyl chamber we have 〈x, α〉 = 0 for some root α, then this point belongs
to a wall. The Weyl chamber consisting of points x such that
〈x, αi〉 ≥ 0, i = 1, 2, . . . , n,
is called the dominant Weyl chamber. It is denoted by D+. Elements of D+ are called dominant.
If 〈x, αi〉 > 0, i = 1, 2, . . . , n, then x is called strictly dominant element.
The set Q of all linear combinations
Q =
{
n∑
i=1
aiαi | ai ∈ Z
}
≡
⊕
i
Zαi
is called a root lattice corresponding to a given Coxeter–Dynkin diagram. Its subset
Q+ =
{
n∑
i=1
aiαi | ai = 0, 1, 2, . . .
}
is called a positive root lattice.
Antisymmetric Orbit Functions 7
To each root α ∈ R there corresponds the coroot α∨ defined by the formula
α∨ =
2α
〈α, α〉
.
It is easy to see that α∨∨ = α. The set Q∨ of all linear combinations
Q∨ =
{
n∑
i=1
aiα
∨
i | ai ∈ Z
}
≡
⊕
i
Zα∨i
is called a coroot lattice corresponding to a given Coxeter–Dynkin diagram. The subset
Q∨
+ =
{
n∑
i=1
aiα
∨
i | ai = 0, 1, 2, . . .
}
is called a positive coroot lattice.
As noted above, the set of simple roots αi, i = 1, 2, . . . , n, form a basis of the space En. In
addition to the α-basis, it is convenient to introduce the so-called ω-basis, ω1, ω2, . . . , ωn (also
called the basis of fundamental weights). The two bases are dual to each other in the following
sense:
2〈αj , ωk〉
〈αj , αj〉
≡ 〈α∨j , ωk〉 = δjk, j, k ∈ {1, 2, . . . , n}. (2.4)
The ω-basis (as well as the α-basis) is not orthogonal.
Note that the factor 2/〈αj , αj〉 can take only three values. Indeed, with the standard nor-
malization of root lengths, we have
2
〈αk, αk〉
= 1 for roots of An, Dn, E6, E7, E8,
2
〈αk, αk〉
= 1 for long roots of Bn, Cn, F4, G2,
2
〈αk, αk〉
= 2 for short roots of Bn, Cn, F4,
2
〈αk, αk〉
= 3 for short roots of G2.
For this reason, we get
α∨k = αk for roots of An, Dn, E6, E7, E8,
α∨k = αk for long roots of Bn, Cn, F4, G2,
α∨k = 2αk for short roots of Bn, Cn, F4,
α∨k = 3αk for short roots of G2.
The α- and ω-bases are related by the Cartan matrix (2.1) and by its inverse:
αj =
n∑
k=1
Mjk ωk, ωj =
n∑
k=1
(M−1)jkαk. (2.5)
For ranks 2 and 3 the inverse Cartan matrices are of the form
A2 :
1
3
(
2 1
1 2
)
, C2 :
(
1 1/2
1 1
)
, G2 :
(
2 3
1 2
)
,
8 A. Klimyk and J. Patera
A3 :
1
4
3 2 1
2 4 2
1 2 3
, B3 :
1
2
2 2 2
2 4 4
1 2 3
, C3 :
1
2
2 2 1
2 4 2
2 4 3
.
Later on we need to calculate scalar products 〈x, y〉 when x and y are given by coordinates xi
and yi in ω-basis. It is given by the formula
〈x, y〉 =
1
2
n∑
j,k=1
xjyk(M−1)jk〈αk |αk〉 = xM−1DyT = xSyT , (2.6)
where D is the diagonal matrix diag (1
2〈α1, α1〉, . . . , 1
2〈αn, αn〉). Matrices S, called ‘quadratic
form matrices’, are shown in [27] for all connected Coxeter–Dynkin diagrams.
The sets P and P+, defined as
P = Zω1 + · · ·+ Zωn ⊃ P+ = Z≥0 ω1 + · · ·+ Z≥0 ωn,
are called respectively the weight lattice and the cone of dominant weights. The set P can be
characterized as a set of all λ ∈ En such that
2〈αj , λ〉
〈αj , αj〉
= 〈α∨j , λ〉 ∈ Z
for all simple roots αj . Clearly, Q ⊂ P . Below we shall need also the set P+
+ of dominant weights
of P+, which do not belong to any Weyl chamber (the set of integral strictly dominant weights).
Then λ ∈ P+
+ means that 〈λ, αi〉 > 0 for all simple roots αi. We have
P+
+ = Z>0ω1 + Z>0ω2 + · · ·+ Z>0ωn.
The smallest dominant weights of P+, different from zero, coincide with the elements ω1, ω2,
. . . , ωn of the ω-basis. They are called fundamental weights. They are highest weights of funda-
mental irreducible representations of the corresponding simple Lie algebra L.
Through the paper we often use the following notation for weights in ω-basis:
z =
n∑
j=1
ajωj = (a1 a2 . . . an), a1, . . . , an ∈ Z.
If x =
n∑
j=1
bjα
∨
j , then
〈z, x〉 =
n∑
j=1
ajbj . (2.7)
2.4 Highest root
There exists a unique highest (long) root ξ and a unique highest short root ξs. The highest
(long) root can be written as
ξ =
n∑
i=1
miαi =
n∑
i=1
mi
〈αi, αi〉
2
α∨i ≡
n∑
i=1
qiα
∨
i . (2.8)
The coefficients mi and qi can be viewed as attached to the i-th node of the diagram. They are
called marks and comarks and are often listed in the literature (see, for example, [27]). In root
Antisymmetric Orbit Functions 9
systems with two lengths of roots, namely in Bn, Cn, F4 and G2, the highest (long) root ξ is of
the form
Bn : ξ = (0 1 0 . . . 0) = α1 + 2α2 + 2α3 + · · ·+ 2αn, (2.9)
Cn : ξ = (2 0 . . . 0) = 2α1 + 2α2 + · · ·+ 2αn−1 + αn, (2.10)
F4 : ξ = (1 0 0 0) = 2α1 + 3α2 + 4α3 + 2α4, (2.11)
G2 : ξ = (1 0) = 2α1 + 3α2. (2.12)
For An, Dn, and En, all roots are of the same length, hence ξs = ξ. We have
An : ξ = (1 0 . . . 0 1) = α1 + α2 + · · ·+ αn, (2.13)
Dn : ξ = (0 1 0 . . . 0) = α1 + 2α2 + · · ·+ 2αn−2 + αn−1 + αn, (2.14)
E6 : ξ = (0 1 0 . . . 0) = α1 + 2α2 + 3α3 + 2α4 + α5 + 2α6, (2.15)
E7 : ξ = (1 0 0 . . . 0) = 2α1 + 3α2 + 4α3 + 3α4 + 2α5 + α6 + 2α7, (2.16)
E8 : ξ = (0 0 . . . 0 1) = 2α1 + 3α2 + 4α3 + 5α4 + 6α5 + 4α6 + 2α7 + 3α8. (2.17)
Note that for highest root ξ we have
ξ∨ = ξ. (2.18)
Moreover, if all simple roots are of the same length, then
α∨i = αi.
For this reason,
(q1, q2, . . . , qn) = (m1,m2, . . . ,mn).
for An, Dn and En. Formulas (2.13)–(2.18) determine these numbers. For short roots αi of Bn,
Cn and F4 we have α∨i = 2αi. For short root α2 of G2 we have α∨2 = 3α2. For this reason,
(q1, q2, . . . , qn) = (1, 2, . . . , 2, 1) for Bn,
(q1, q2, . . . , qn) = (1, 1, . . . , 1, 1) for Cn,
(q1, q2, q3, q4) = (2, 3, 2, 1) for F4,
(q1, q2) = (2, 1) for G2.
To each root system R there corresponds an extended root system (which is also called an
affine root system). It is constructed with the help of the highest root ξ of R. Namely, if
α1, α2, . . . , αn is a set of all simple roots, then the roots
α0 := −ξ, α1, α2, . . . , αn
constitute a set of simple roots of the corresponding extended root system. Taking into account
the orthogonality (non-orthogonality) of the root α0 to other simple roots, the diagram of an
extended root system can be constructed (which is an extension of the corresponding Coxeter–
Dynkin diagram; see, for example, [28]). Note that for all simple Lie algebras (except for An)
only one simple root is orthogonal to the root α0. In the case of An, the two simple roots α1
and αn are not orthogonal to α0.
10 A. Klimyk and J. Patera
2.5 Affine Weyl groups
We are interested in antisymmetric orbit functions which are given on the Euclidean space En.
These functions are anti-invariant with respect to action by elements of a Weyl groupW , which is
a transformation group of En. However, W does not describe all symmetries of orbit functions
corresponding to weights λ ∈ P+
+ . A whole group of anti-invariances of antisymmetric orbit
functions is isomorphic to the affine Weyl groupW aff which is an extension of the Weyl groupW .
This group is defined as follows.
Let α1, α2, . . . , αn be simple roots in the Euclidean space En and let W be the corresponding
Weyl group. The group W is generated by reflections rαi , i = 1, 2, . . . , n. In order to construct
the affine Weyl group W aff , corresponding to the group W , we have to add an additional
reflection. This reflection is constructed as follows.
We consider the reflection rξ with respect to the (n− 1)-dimensional subspace (hyperplane)
Xn−1 containing the origin and orthogonal to the highest (long) root ξ, given in (2.8):
rξx = x− 2〈x, ξ〉
〈ξ, ξ〉
ξ. (2.19)
Clearly, rξ ∈ W . We shift the hyperplane Xn−1 by the vector ξ∨/2, where ξ∨ = 2ξ/〈ξ, ξ〉.
(Note that by (2.18) we have ξ∨ = ξ. However, it is convenient to use here ξ∨.) The reflection
with respect to the hyperplane Xn−1 + ξ∨/2 will be denoted by r0. Then in order to fulfill the
transformation r0 we have to fulfill the transformation rξ and then to shift the result by ξ∨,
that is,
r0x = rξx+ ξ∨.
We have r00 = ξ∨ and it follows from (2.19) that r0 maps x+ ξ∨/2 to
rξ(x+ ξ∨/2) + ξ∨ = x+ ξ∨/2− 〈x, ξ∨〉ξ.
Therefore,
r0(x+ ξ∨/2) = x+ ξ∨/2− 2〈x, ξ〉
〈ξ, ξ〉
ξ = x+ ξ∨/2− 2〈x, ξ∨〉
〈ξ∨, ξ∨〉
ξ∨
= x+ ξ∨/2− 2〈x+ ξ∨/2, ξ∨〉
〈ξ∨, ξ∨〉
ξ∨ +
2〈ξ∨/2, ξ∨〉
〈ξ∨, ξ∨〉
ξ∨.
Denoting x+ ξ∨/2 by y we obtain that r0 is given also by the formula
r0y = y +
(
1− 2〈y, ξ∨〉
〈ξ∨, ξ∨〉
)
ξ∨ = ξ∨ + rξy. (2.20)
The element r0 does not belong to W since elements of W do not move the point 0 ∈ En.
The hyperplane Xn−1 + ξ∨/2 coincides with the set of points y such that r0y = y. It follows
from (2.20) that this hyperplane is given by the equation
1 =
2〈y, ξ∨〉
〈ξ∨, ξ∨〉
= 〈y, ξ〉 =
n∑
k=1
akqk, (2.21)
where
y =
n∑
k=1
akωk, ξ =
n∑
k=1
qkα
∨
k
(see (2.7)).
Antisymmetric Orbit Functions 11
A group of transformations of the Euclidean space En generated by reflections r0, rα1 , . . . , rαn
is called the affine Weyl group of the root system R and is denoted by W aff or by W aff
R (if is
necessary to indicate the initial root system), see [28].
Adjoining the reflection r0 to the Weyl group W completely change properties of the group
W aff .
If rξ is the reflection with respect to the hyperplane Xn−1, then due to (2.19) and (2.20) for
any x ∈ En we have
r0rξx = r0(rξx) = ξ∨ + rξrξx = x+ ξ∨.
Clearly, (r0rξ)kx = x + kξ∨, k = 0,±1,±2, . . . , that is, the set of elements (r0rξ)k, k =
0,±1,±2, . . . , is an infinite commutative subgroup of W aff . This means that (unlike to the
Weyl group W ) W aff is an infinite group.
Since r00 = ξ∨, for any w ∈W we have
wr00 = wξ∨ = ξ∨w,
where ξ∨w is a coroot of the same length as the coroot ξ∨. For this reason, wr0 is the reflection
with respect to the (n−1)-hyperplane perpendicular to the root ξ∨w and containing the point ξ∨w/2.
Moreover,
(wr0)rξ∨wx = x+ ξ∨w.
We also have ((wr0)rξ∨w)kx = x + kξ∨w, k = 0,±1,±2, . . . . Since w is any element of W , then
the set wξ∨, w ∈ W , coincides with the set of coroots of R∨, corresponding to all long roots of
the root system R. Thus, the set W aff · 0 coincides with the lattice Q∨
l generated by coroots α∨
taken for all long roots α from R.
It is checked for each type of root systems that each coroot ξ∨s for a short root ξs of R is
a linear combination of coroots wξ∨ ≡ ξw, w ∈W , with integral coefficients, that is, Q∨ = Q∨
l .
Therefore, The set W aff · 0 coincides with the coroot lattice Q∨ of R.
Let Q̂∨ be the subgroup of W aff generated by the elements
(wr0)rw, w ∈W, (2.22)
where rw ≡ rξ∨w for w ∈W . Since elements (2.22) pairwise commute with each other (since they
are shifts), Q̂∨ is a commutative group. The subgroup Q̂∨ can be identified with the coroot
lattice Q∨. Namely, if for g ∈ Q̂∨ we have g · 0 = γ ∈ Q∨, then g is identified with γ. This
correspondence is one-to-one.
The subgroups W and Q̂∨ generate W aff since a subgroup of W aff , generated by W and Q̂∨,
contains the element r0. The group W aff is a semidirect product of its subgroups W and Q̂∨,
where Q̂∨ is an invariant subgroup (see Section 5.2 in [1] for details).
2.6 Fundamental domain
An open connected simply connected set D ⊂ En is called a fundamental domain for the group
W aff (for the group W ) if it does not contains equivalent points (that is, points x and x′ such
that x = wx) and if its closure contains at least one point from each W aff -orbit (from each
W -orbit). It is evident that the dominant Weyl chamber (without walls of this chamber) is
a fundamental domain for the Weyl group W . Recall that this domain consists of all points
x = a1ω1 + · · ·+ anωn ∈ En for which
ai = 〈x, α∨i 〉 > 0, i = 1, 2, . . . , n.
12 A. Klimyk and J. Patera
We wish to describe a fundamental domain for the group W aff . Since W ⊂ W aff , it can be
chosen as a subset of the dominant Weyl chamber for W .
We have seen that the element r0 ∈ W aff is a reflection with respect to the hyperplane
Xn−1 + ξ∨/2, orthogonal to the root ξ and containing the point ξ∨/2. This hyperplane is given
by the equation (2.21). This equation shows that the hyperplane Xn−1 + ξ∨/2 intersects the
axices, determined by the vectors ωi, in the points ωi/qi, i = 1, 2, . . . , n, where qi are such as
in (2.21). We create the simplex with n+ 1 vertices in the points
0,
ω1
q1
, . . . ,
ωn
qn
. (2.23)
By the definition of this simplex and by (2.21), this simplex consists of all points y of the
dominant Weyl chamber for which 〈y, ξ〉 ≤ 1. Clearly, the interior F of this simplex belongs to
the dominant Weyl chamber. The following theorem is true (see, for example, [1]):
Theorem 1. The set F is a fundamental domain for the affine Weyl group W aff .
For the rank 2 cases the fundamental domain is the interior of the simplex with the following
vertices:
A2 : {0, ω1, ω2},
C2 : {0, ω1, ω2},
G2 : {0, ω1
2 , ω2}.
3 Weyl group signed orbits
3.1 Signed orbits
As we have seen, the (n − 1)-dimensional linear subspaces Xα of En, orthogonal to positive
roots α and containing the origin, divide the space En into connected parts, which are called Weyl
chambers. A number of such chambers is equal to an order of the corresponding Weyl group W .
Elements of the Weyl group permute these chambers. There exists a single chamber D+ such
that 〈αi, x〉 ≥ 0, x ∈ D+, i = 1, 2, . . . , n. It is the dominant Weyl chamber.
Clearly, the cone of dominant weights P+ belongs to the dominant Weyl chamber D+. (Note
that it is not a case for the set Q+.) We have P ∩D+ = P+.
Let y be an arbitrary dominant element of the Euclidean space En, which does not lie on
some Weyl chamber. We act upon y by all elements of the Weyl group W . As a result, we
obtain a set of elements wy, w ∈ W , which is called Weyl group orbit. All these elements are
pairwise different. We attach to each point wy a sign coinciding with a sign of detw. The set
of all points wy, w ∈ W , together with their signs is called a signed orbit of the point y with
respect the Weyl group (or a Weyl group signed orbit, containing y). Points of signed orbits
will be denoted by x+ or x−, depending on a sign. Sometimes, we denote points wy, y ∈ D+, of
the signed orbit, containing the point y, as wydetw, where instead of a sign we have +1 or −1,
respectively.
An orbit (where points do not have signs) of a point y ∈ D+ is denoted by O(y) or OW (y).
A size of an orbit O(y) is a number |O(y)| of its elements. Each Weyl chamber contains only
one point of a fixed orbit Q(y). A signed orbit of a strictly dominant point y ∈ En is denoted
by O±(y) or O±
W (y).
Note that orbits O(y) are defined for any dominant elements y ∈ En. Signed orbits O±(y)
can be defined only for strictly dominant y ∈ En.
Antisymmetric Orbit Functions 13
3.2 Signed orbits of A1, A1 × A1, A2, C2, G2
Assuming that a > 0 and b > 0, we list the contents of signed orbits in ω-basis:
A1 : O±(a) 3 (a)+, (−a)− (3.1)
A1 ×A1 : O±(a b) 3 (a b)+, (−a b)−, (a −b)−, (−a −b)+ (3.2)
A2 : O±(a b) 3 (a b)+, (−a a+b)−, (a+b −b)−,
(−b −a)−, (−a−b a)+, (b −a−b)+. (3.3)
In the cases of C2 and G2 (where the second simple root is the longer one for C2 and the shorter
one for G2) we have
C2 : O±(a b) 3 (a b)+, (−a a+b)−, (a+2b −b)−, (a+2b −a−b)+,
(−a −b)+, (−a −a−b)−, (−a−2b b)−, (−a−2b a+b)+, (3.4)
G2 : O±(a b) 3 ±(a b)+, ±(−a 3a+b)−, ±(a+b −b)−,
± (2a+b −3a−b)+, ±(−a−b 3a+2b)+, ±(−2a−b 3a+2b)−, (3.5)
where ±(c, d)+ means two signed points (c, d)+ and (−c,−d)+.
As we see, for each point (c d) of a signed orbit of C2 or G2 there exists in the orbit the point
(−c −d) with the same sign.
3.3 The case of An
In the cases of Coxeter–Dynkin diagrams An−1, Bn, Cn, Dn, root and weight lattices, Weyl
groups and signed orbits are described in a simple way by using the orthogonal coordinate
system in En. In particular, this coordinate system is useful under practical work with signed
orbits.
In the case An it is convenient to describe root and weight lattices, Weyl group and antisym-
metric orbit functions in the subspace of the Euclidean space En+1, given by the equation
x1 + x2 + · · ·+ xn+1 = 0,
where x1, x2, . . . , xn+1 are orthogonal coordinates of a point x ∈ En+1. The unit vectors in
directions of these coordinates are denoted by ej , respectively. Clearly, ei⊥ej , i 6= j. The set of
roots of An is given by the vectors
αij = ei − ej , i 6= j.
The roots
αij = ei − ej , i < j,
are positive and the roots
αi ≡ αi,i+1 = ei − ei+1, i = 1, 2, . . . , n,
constitute the system of simple roots.
If x =
n+1∑
i=1
xiei, x1 + x2 + · · · + xn+1 = 0, is a point of En+1, then this point belongs to the
dominant Weyl chamber D+ if and only if
x1 ≥ x2 ≥ · · · ≥ xn+1.
14 A. Klimyk and J. Patera
Indeed, if this condition is fulfilled, then 〈x, αi〉 = xi − xi+1 ≥ 0, i = 1, 2, . . . , n, and x is
dominant. Conversely, if x is dominant, then 〈x, αi〉 ≥ 0 and this condition is fulfilled. The
point x is strictly dominant if and only if
x1 > x2 > · · · > xn+1.
If λ =
n∑
i=1
λiωi, then the coordinates λi in the ω-coordinates are connected with the orthogonal
coordinates mj of λ =
n+1∑
i=1
miei by the formulas
m1 =
n
n+ 1
λ1 +
n− 1
n+ 1
λ2 +
n− 2
n+ 1
λ3 + · · ·+ 2
n+ 1
λn−1 +
1
n+ 1
λn,
m2 = − 1
n+ 1
λ1 +
n− 1
n+ 1
λ2 +
n− 2
n+ 1
λ3 + · · ·+ 2
n+ 1
λn−1 +
1
n+ 1
λn,
m3 = − 1
n+ 1
λ1 −
2
n+ 1
λ2 +
n− 2
n+ 1
λ3 + · · ·+ 2
n+ 1
λn−1 +
1
n+ 1
λn,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
mn = − 1
n+ 1
λ1 −
2
n+ 1
λ2 −
3
n+ 1
λ3 − · · · − n− 1
n+ 1
λn−1 +
1
n+ 1
λn,
mn+1 = − 1
n+ 1
λ1 −
2
n+ 1
λ2 −
3
n+ 1
λ3 − · · · − n− 1
n+ 1
λn−1 −
n
n+ 1
λn.
The inverse formulas are
λi = mi −mi+1, i = 1, 2, . . . , n. (3.6)
By means of the formula
rαλ = λ− 2〈λ, α〉
〈α, α〉
α (3.7)
for the reflection with respect to the hyperplane, orthogonal to a root α, we can find that the
reflection rαij acts upon the vector λ =
n+1∑
i=1
miei, given by orthogonal coordinates, by permuting
the coordinates mi and mj . Since for each i and j, 1 ≤ i, j ≤ n+ 1, there exists a root αij , the
Weyl group W (An) consists of all permutations of the orthogonal coordinates m1,m2, . . . ,mn+1
of a point λ, that is, W (An) coincides with the symmetric group Sn+1.
Sometimes (for example, if we wish that coordinates would be integers or non-negative inte-
gers), it is convenient to introduce orthogonal coordinates x1, x2, . . . , xn+1 for An in such a way
that
x1 + x2 + · · ·+ xn+1 = m,
where m is some fixed real number. They are obtained from the previous orthogonal coordinates
by adding the same number m/(n + 1) to each coordinate. Then, as one can see from (3.6),
ω-coordinates λi = xi − xi+1 and the Weyl group W do not change. Sometimes, it is natural to
use orthogonal coordinates x1, x2, . . . , xn+1 for which all xi are non-negative.
We need below the half-sum ρ of the positive roots of An, ρ = 1
2
∑
α>0
α. It is easy to see that
up to a common constant we have
ρ = ne1 + (n− 1)e2 + · · ·+ en,
Antisymmetric Orbit Functions 15
that is, in orthogonal coordinates we have
ρ = (n, n− 1, . . . , 1, 0). (3.8)
In the non-orthogonal ω-coordinates we have
ρ = ω1 + ω2 + · · ·+ ωn.
The signed orbit O±(λ), λ = (m1,m2, . . . ,mn+1), m1 > m2 > · · · > mn+1, consists of all
points
(mi1 ,mi2 , . . . ,min+1)
sgn (detw)
obtained from (m1,m2, . . . ,mn+1) by permutations w ∈W ≡ Sn+1. Below instead of sgn (detw)
we write simply detw.
3.4 The case of Bn
Orthogonal coordinates of a point x ∈ En are denoted by x1, x2, . . . , xn. We denote by ei the
corresponding unit vectors. Then the set of roots of Bn is given by the vectors
α±i,±j = ±ei ± ej , i 6= j, α±i = ±ei, i = 1, 2, . . . , n
(all combinations of signs must be taken). The roots
αi,±j = ei ± ej , i < j, αi = ei, i = 1, 2, . . . , n,
are positive and n roots
αi := ei − ei+1, i = 1, 2, . . . , n− 1, αn = en
constitute the system of simple roots.
It is easy to see that if λ =
n∑
i=1
miei is a point of En, then this point belongs to the dominant
Weyl chamber D+ if and only if
m1 ≥ m2 ≥ · · · ≥ mn ≥ 0.
Moreover, this point is strictly dominant if and only if
m1 > m2 > · · · > mn > 0.
If λ =
n∑
i=1
λiωi, then the coordinates λi in the ω-coordinates are connected with the coordi-
nates mj of λ =
n∑
i=1
miei by the formulas
m1 =λ1+λ2+· · ·+λn−1+1
2λn,
m2 = λ2+· · ·+λn−1+1
2λn,
· · · · · · · · · · · · · · · · · ·
mn = 1
2λn,
The inverse formulas are
λi = mi −mi+1, i = 1, 2, . . . , n− 1, λn = 2mn.
16 A. Klimyk and J. Patera
It is easy to see that if λ ∈ P+, then the coordinates m1,m1, . . . ,mn are all integers or all
half-integers.
The half-sum ρ of positive roots of Bn, ρ = 1
2
∑
α>0
α, in orthogonal coordinates has the form
ρ = (n− 1
2 , n−
3
2 , . . . ,
1
2). (3.9)
In ω-coordinates we have ρ = ω1 + ω2 + · · ·+ ωn.
By means of the formula (3.7) we find that the reflection rα acts upon orthogonal coordinates
of the vector λ =
n∑
i=1
miei by permuting i-th and j-th coordinates if α = ±(ei − ej), as the
permutation of i-th and j-th coordinates and the change of their signs if α = ±(ei + ej), and
as the change of a sign of i-th coordinate if α = ±ei. Thus, the Weyl group W (Bn) consists of
all permutations of the orthogonal coordinates m1,m2, . . . ,mn of a point λ with possible sign
alternations of any number of them.
The signed orbit O±(λ), λ = (m1,m2, . . . ,mn), m1 > m2 > · · · > mn > 0, consists of all
points
(±mi1 ,±mi2 , . . . ,±min)detw (3.10)
(each combination of signs is possible) obtained from (m1,m2, . . . ,mn) by permutations and
alternations of signs which constitute an element w of the Weyl group W (Bn). Moreover,
detw is equal to ±1 depending on whether w consists of even or odd number of reflections
and alternations of signs. A sign of detw can be determined as follows. We represent w as
a product w = εs, where s is a permutation of (m1,m2, . . . ,mn) and ε is an alternation of
signs of coordinates. Then detw = (det s)εi1εi2 · · · εin , where det s is defined as in the previous
subsection and εij is a sign of ij-th coordinate.
3.5 The case of Cn
In the orthogonal system of coordinates of the Euclidean space En the set of roots of Cn is given
by the vectors
α±i,±j = ±ei ± ej , i 6= j, α±i = ±2ei, i = 1, 2, . . . , n,
where ei is the unit vector in the direction of i-th coordinate xi (all combinations of signs must
be taken). The roots
αi,±j = ei ± ej , i < j, αi = 2ei, i = 1, 2, . . . , n,
are positive and n roots
αi := ei − ei+1, i = 1, 2, . . . , n− 1, αn = 2en
constitute the system of simple roots.
It is easy to see that a point λ =
n∑
i=1
miei ∈ En belongs to the dominant Weyl chamber D+
if and only if
m1 ≥ m2 ≥ · · · ≥ mn ≥ 0.
This point is strictly dominant if and only if
m1 > m2 > · · · > mn > 0.
Antisymmetric Orbit Functions 17
If λ =
n∑
i=1
λiωi, then the coordinates λi in the ω-coordinates are connected with the coordi-
nates mj of λ =
n∑
i=1
miei by the formulas
m1 =λ1+λ2+· · ·+λn−1+λn,
m2 = λ2+· · ·+λn−1+λn,
· · · · · · · · · · · · · · · · · ·
mn = λn.
The inverse formulas are
λi = mi −mi+1, i = 1, 2, . . . , n− 1, λn = mn.
If λ ∈ P+, then all coordinates mi are integers.
The half-sum ρ of positive roots of Cn, ρ = 1
2
∑
α>0
α, in orthogonal coordinates has the form
ρ = (n, n− 1, . . . , 2, 1). (3.11)
By means of the formula (3.7) we find that the reflection rα acts upon orthogonal coordinates
of the vector λ =
n∑
i=1
miei by permuting i-th and j-th coordinates if α = ±(ei − ej), as the
permutation of i-th and j-th coordinates and the change of their signs if α = ±(ei + ej), and as
the change of a sign of i-th coordinate if α = ±2ei. Thus, the Weyl group W (Cn) consists of all
permutations of the orthogonal coordinates m1,m2, . . . ,mn of a point λ with sign alternations
of some of them, that is, this Weyl group acts on orthogonal coordinates exactly in the same
way as the Weyl group W (Bn) does.
The signed orbit O±(λ), λ = (m1,m2, . . . ,mn), m1 > m2 > · · · > mn > 0, consists of all
points
(±mi1 ,±mi2 , . . . ,±min+1)
detw
(each combination of signs is possible) obtained from (m1,m2, . . . ,mn) by permutations and
alternations of signs which constitute an element w of the Weyl group W (Cn). Moreover, detw
is equal to ±1 depending on whether w consists of even or odd numbers of reflections and
alternations of signs. Since W (Cn) = W (Bn), then a sign of detw is determined as in the
case Bn.
As we see, in the orthogonal coordinates signed orbits for Cn coincide with signed orbits
of Bn.
3.6 The case of Dn
In the orthogonal system of coordinates of the Euclidean space En the set of roots of Dn is given
by the vectors
α±i,±j = ±ei ± ej , i 6= j,
where ei is the unit vector in the direction of i-th coordinate (all combinations of signs must be
taken). The roots
αi,±j = ei ± ej , i < j,
18 A. Klimyk and J. Patera
are positive and n roots
αi := ei − ei+1, i = 1, 2, . . . , n− 1, αn = en−1 + en
constitute the system of simple roots.
It is easy to see that if λ =
n∑
i=1
miei is a point of En, then this point belongs to the dominant
Weyl chamber D+ if and only if
m1 ≥ m2 ≥ · · · ≥ mn−1 ≥ |mn|.
This point is strictly dominant if and only if
m1 > m2 > · · · > mn−1 > |mn|
(in particular, mn can take the value 0).
If λ =
n∑
i=1
λiωi, then the coordinates λi in the ω-coordinates are connected with the coordi-
nates mj of λ =
n∑
i=1
miei by the formulas
m1 =λ1+λ2+· · ·+λn−2+1
2(λn−1+λn),
m2 = λ2+· · ·+λn−2+1
2(λn−1+λn),
· · · · · · · · · · · · · · · · · · · · ·
mn−1 = 1
2(λn−1+λn),
mn = 1
2(λn−1−λn),
The inverse formulas are
λi = mi −mi+1, i = 1, 2, . . . , n− 2, λn−1 = mn−1 +mn, λn = mn−1 −mn.
If λ ∈ P+, then the coordinates m1,m2, . . . ,mn are all integers or all half-integers.
The half-sum ρ of positive roots of Dn, ρ = 1
2
∑
α>0
α, in orthogonal coordinates has the form
ρ = (n− 1, n− 2, . . . , 1, 0). (3.12)
By means of the formula (3.7) for the reflection rα we find that rα acts upon orthogonal
coordinates of the vector λ =
n∑
i=1
miei by permuting i-th and j-th coordinates if α = ±(ei−ej),
and as the permutation of i-th and j-th coordinates and the change of their signs if α = ±(ei +
ej). Thus, the Weyl group W (Dn) consists of all permutations of the orthogonal coordinates
m1,m2, . . . ,mn of a point λ with sign alternations of even number of them.
Since an alternation of signs of two coordinates xi and xj is a product of two reflections rα
with α = (ei + ej) and with α = (ei − ej), a sign of the determinant of this alternation is plus.
Note that |W (Dn)| = 1
2 |W (Bn)|.
The signed orbit O±(λ), λ = (m1,m2, . . . ,mn), m1 > m2 > · · · > mn > 0, consists of all
points
(±mi1 ,±mi2 , . . . ,±min+1)
detw
obtained from (m1,m2, . . . ,mn) by permutations and alternations of even number of signs which
constitute an element w of the Weyl group W (Dn). Moreover, detw is equal to ±1 and a sign
of detw is determined as follows. The element w ∈ W (Dn) can be represented as a product
w = τs, where s is a permutation from Sn and τ is an alternation of even number of coordinates.
Then detw = det s. Indeed, a determinant of a transform, given by an element of W which is
an alternation of two signs, is equal to +1 (since this element can be represented as a product
of two reflections).
Antisymmetric Orbit Functions 19
3.7 Signed orbits of A3
Signed orbits for A3, B3 and C3 can be calculated by using the orthogonal coordinates in
the corresponding Euclidean space, described above, and the description of the Weyl groups
W (A3), W (B3) and W (C3) in the orthogonal coordinate systems. Below we give results of
such calculations. Points λ of signed orbits are given in the ω-coordinates as (a b c), where
λ = aω1 + bω2 + cω3.
The signed orbit O±(a b c), a > 0, b > 0, c > 0, of A3 contains the points
O±(a b c) 3 (a b c)+, (a+b −b b+c)−, (a+b c −b−c)+, (a b+c −c)−,
(a+b+c −c −b)−, (a+b+c −b−c b)+, (−a a+b c)−, (−a a+b+c −c)+,
(b −a−b a+b+c)+, (b+c −a−b−c a+b)−, (−a−b a b+c)+, (−b −a a+b+c)−
and the points, contragredient to these points, where the contragredient of the point (a′ b′ c′)+
is (−c′ −b′ −a′)+ and the contragredient of the point (a′ b′ c′)− is (−c′ −b′ −a′)−.
3.8 Signed orbits of B3
As in the previous case, points λ of signed orbits are given by the ω-coordinates (a b c), where
λ = aω1 + bω2 + cω3. The signed orbit O±(a b c), a > 0, b > 0, c > 0, of B3 contains the points
O±(a b c) 3 (a b c)+, (a+b −b 2b+c)−, (−a a+b c)−, (b −a−b 2a+2b+c)+,
(−a−b a 2b+c)+, (−b −a 2a+2b+c)−, (a b+c −c)−, (a+b+c −b−c 2b+c)+,
(−a a+b+c −c)+, (b+c −a−b−c 2a+2b+c)−, (−a−b−c a 2b+c)−,
(−b−c −a 2a+2b+c)+, (−a−2b−c b c)−, (−a−b−c −b 2b+c)+,
(a+2b+c −a−b−c c)+, (b a+b+c −2a−2b−c)−, (a+b+c −a−2b−c 2b+c)−,
(−b a+2b+c −2a−2b−c)+, (−a−2b−c b+c −c)+, (−a−b −b−c 2b+c)−,
(a+2b+c −a−b −c)−, (b+c a+b −2a−2b−c)+, (a+b −a−2b−c 2b+c)+,
(−b−c a+2b+c −2a−2b−c)−
and also all these points taken with opposite signs of coordinates, signs of these points are also
opposite.
3.9 Signed orbits of C3
As in the previous cases, points λ of signed orbits are given by the ω-coordinates (a b c), where
λ = aω1 + bω2 + cω3. The signed orbit O±(a b c), a > 0, b > 0, c > 0, of C3 contains the points
O±(a b c) 3 (a b c)+, (a+b −b b+c)−, (−a a+b c)−, (b −a−b a+b+c)+,
(−a−b a b+c)+, (−b −a a+b+c)−, (a b+2c −c)−, (a+b+2c −b−2c b+c)+,
(−a a+b+2c −c)+, (b+2c −a−b−2c a+b+c)−, (−a−b−2c a b+c)−,
(−b−2c −a a+b+c)+, (−a−2b−2c b c)−, (−a−b−2c −b b+c)+,
(a+2b+2c −a−b−2c c)+, (b a+b+2c −a−b−c)−, (a+b+2c −a−2b−2c b+c)−,
(−b a+2b+2c −a−b−c)+, (−a−2b−2c b+2c −c)+, (−a−b −b−2c b+c)−,
(a+2b+2c −a−b −c)−, (b+2c a+b −a−b−c)+, (a+b −a−2b−2c b+c)+,
(−b−2c a+2b+2c −a−b−c)−
and also all these points taken with opposite signs of coordinates, signs of these points are also
opposite.
20 A. Klimyk and J. Patera
4 Antisymmetric orbit functions
4.1 Definition
The exponential functions e2πi〈m,x〉, x ∈ En, with fixed m = (m1,m2, . . . ,mn) determine the
Fourier transform on En. Antisymmetric orbit functions are an antisymmetrized (with respect
to a Weyl group) version of exponential functions. Correspondingly, they determine an anti-
symmetrized version of the Fourier transform.
First we define symmetric orbit functions, studied in [1]. Let W be a Weyl group of transfor-
mations of the Euclidean space En. To each element λ ∈ En from the dominant Weyl chamber
(that is, 〈λ, αi〉 ≥ 0 for all simple roots αi) there corresponds a symmetric orbit function φλ
on En, which is given by the formula
φλ(x) =
∑
µ∈O(λ)
e2πi〈µ,x〉, x ∈ En, (4.1)
where O(λ) is the W -orbit of the element λ. The number of summands is equal to the size
|O(λ)| of the orbit O(λ) and we have φλ(0) = |O(λ)|.
Sometimes (see, for example, [14] and [15]), it is convenient to use a modified definition of
orbit functions:
φ̂λ(x) = |Wλ|φλ(x), (4.2)
where Wλ is a subgroup in W whose elements leave λ fixed. Then for all orbit functions φ̂λ we
have φ̂λ(0) = |W |.
Antisymmetric orbit functions are defined (see [22] and [29]) for dominant elements λ, which
do not belong to a wall of the dominant Weyl chamber (that is, for strictly dominant elements λ).
The antisymmetric orbit function, corresponding to such an element, is defined as
ϕλ(x) =
∑
w∈W
(detw)e2πi〈wλ,x〉, x ∈ En. (4.3)
A number of summands in (4.3) is equal to the size |W | of the Weyl group W . We have
ϕλ(0) = 0.
Symmetric orbit functions φλ for which λ ∈ P+ and antisymmetric orbit functions ϕλ(x) for
which λ ∈ P+
+ are of special interest for representation theory.
Example. Antisymmetric orbit functions for A1. In this case, there exists only one simple
(positive) root α. We have 〈α, α〉 = 2. Then the relation 2〈ω, α〉/〈α, α〉 = 1 means that
〈ω, α〉 = 1. This means that ω = α/2 and 〈ω, ω〉 = 1/2. Elements of P+
+ coincide with mω,
m ∈ Z+. We identify points x of E1 ≡ R with θω. Since the Weyl group W (A1) consists of two
elements 1 and rα, and
rαx = x− 2〈θω, α〉
〈α, α〉
α = x− θα = x− 2x = −x,
antisymmetric orbit functions ϕλ(x), λ = mω, m > 0, are given by the formula
ϕλ(x) = e2πi〈mω,θω〉 − e−2πi〈mω,θω〉 = eπimθ − e−πimθ = 2i sin(πmθ).
Note that for the symmetric orbit function φλ(x) we have φλ(x) = 2 cos(πmθ).
Antisymmetric Orbit Functions 21
4.2 Antisymmetric orbit functions of A2
Antisymmetric orbit functions for the Coxeter–Dynkin diagrams of rank 2 were given in [22]. In
this subsection we give these functions for A2.
Put λ = aω1 + bω2 ≡ (a b) with a > 0, b > 0. Then for ϕλ(x) ≡ ϕ(a b)(x) we have from (4.3)
that
ϕ(a b)(x) = e2πi〈(a b),x〉 − e2πi〈(−a a+b),x〉 − e2πi〈(a+b −b),x〉
+ e2πi〈(b −a−b),x〉 + e2πi〈(−a−b a),x〉 − e2πi〈(−b −a),x〉.
Using the representation x = ψ1α1 + ψ2α2, one obtains
ϕ(a b)(x) = e2πi(aψ1+bψ2) − e2πi(−aψ1+(a+b)ψ2) − e2πi((a+b)ψ1−bψ2)
+ e2πi(bψ1−(a+b)ψ2) + e2πi((−a−b)ψ1+aψ2) − e2πi(−bψ1−aψ2). (4.4)
The actual expression for ϕ(a b)(x) depends on a choice of coordinate systems for λ and x.
Setting x = θ1ω1 + θ2ω2 and λ as before, we get
ϕ(a b)(x) = e
2πi
3 ((2a+b)θ1+(a+2b)θ2) − e
2πi
3 ((−a+b)θ1+(a+2b)θ2)
− e
2πi
3 ((2a+b)θ1+(a−b)θ2) + e−
2πi
3 ((a−b)θ1+(2a+b)θ2) (4.5)
+ e−
2πi
3 ((a+2b)θ1+(−a+b)θ2) − e−
2πi
3 ((a+2b)θ1+(2a+b)θ2).
Note that φ(a a)(x) are pure imaginary for all a > 0 and
ϕ(a a)(x) = 2i {sin 2πa(ψ1 + ψ2) + sin 2πa(ψ1 − 2ψ2)− sin 2πa(2ψ1 − ψ2)}
= 2i {sin 2πa(θ1 + θ2)− sin 2πaθ1 − sin 2πaθ2} . (4.6)
The pairs ϕ(a b)(x) + ϕ(b a)(x) are always pure imaginary functions.
4.3 Antisymmetric orbit functions of C2 and G2
Putting again λ = aω1 + bω2 = (a b), x = θ1ω1 + θ2ω2 and using the matrices S from (2.6),
which are of the form
S(C2) = 1
2
(
1 1
1 2
)
, S(G2) = 1
6
(
6 3
3 2
)
,
we find the orbit functions for C2 and G2:
C2 : ϕ(a b)(x) = 2 cosπ((a+ b)θ1 + (a+ 2b)θ2)− 2 cosπ(bθ1 + (a+ 2b)θ2)
− 2 cosπ((a+ b)θ1 + aθ2) + 2 cosπ(bθ1 − aθ2), (4.7)
G2 : ϕ(a b)(x) = 2 cosπ((2a+ b)θ1 + (a+ 2
3b)θ2)− 2 cosπ((a+ b)θ1 + (a+ 2
3b)θ2)
− 2 cosπ((2a+ b)θ1 + (a+ 1
3b)θ2) + 2 cosπ((a+ b)θ1 + 1
3bθ2)
+ 2 cosπ(aθ1 + (a+ 1
3b)θ2)− 2 cosπ(aθ1 − 1
3bθ2). (4.8)
As we see, orbit functions for C2 and G2 are real.
22 A. Klimyk and J. Patera
4.4 Antisymmetric orbit functions of An
It is difficult to write down an explicit form of orbit functions for An, Bn, Cn and Dn in
coordinates with respect to the ω- or α-bases. For this reason, for these cases we use the
orthogonal coordinate systems, described in Section 3.
Let λ = (m1,m2, . . . ,mn+1) be a strictly dominant element for An in orthogonal coordinates
described in Subsection 3.3. Then m1 > m2 > · · · > mn+1. The Weyl group in this case
coincides with the symmetric group Sn+1. Then the signed orbit O±(λ) consists of points
(wλ)detw, w ∈ W ≡ Sn+1. Representing points x ∈ En+1 in the orthogonal coordinate system,
x = (x1, x2, . . . , xn+1), and using formula (4.3) we find that
ϕλ(x) =
∑
w∈Sn+1
(detw)e2πi〈w(m1,...,mn+1),(x1,...,xn+1)〉
=
∑
w∈Sn+1
(detw)e2πi((wλ)1x1+···+(wλ)n+1xn+1), (4.9)
where ((wλ)1, (wλ)2, . . . , (wλ)n+1) are the coordinates of the point wλ.
Note that the element −(mn+1,mn, . . . ,m1) is strictly dominant if the element (m1,m2, . . . ,
mn+1) is strictly dominant. In the Weyl group W (An) there exists an element w0 such that
w0(m1,m2, . . . ,mn+1) = (mn+1,mn, . . . ,m1).
Moreover, we have
detw0 = 1 for A4k−1 and A4k,
detw0 = −1 for A4k+1 and A4k+2.
It follows from here that in the expressions for the orbit functions ϕ(m1,m2,...,mn+1)(x) and
ϕ−(mn+1,mn,··· ,m1)(x) there are summands
e2πi〈w0λ,x〉 = e2πi(mn+1x1+···+m1xn+1) and e−2πi(mn+1x1+···+m1xn+1), (4.10)
respectively, which are complex conjugate to each other. Moreover, the first expression is con-
tained with the sign (detw0) in ϕ(m1,m2,...,mn+1)(x), that is, the expressions (4.10) are contained
in ϕ(m1,m2,...,mn+1)(x) and ϕ−(mn+1,mn,...,m1)(x) with the same sign for n = 4k − 1, 4k and with
opposite signs for n = 4k + 1, 4k + 2, k ∈ Z+.
Similarly, in the expressions (4.9) for the function ϕ(m1,m2,...,mn+1)(x) and for the function
ϕ−(mn+1,mn,...,m1)(x) all other summands are (up to a sign, which depends on a value of n)
pairwise complex conjugate. Therefore,
ϕ(m1,m2,...,mn+1)(x) = ϕ−(mn+1,mn,...,m1)(x) (4.11)
for n = 4k − 1, 4k and
ϕ(m1,m2,...,mn+1)(x) = −ϕ−(mn+1,mn,...,m1)(x) (4.12)
for n = 4k + 1, 4k + 2.
If we use for λ the coordinates λi = 〈λ, α∨i 〉 in the ω-basis instead of the orthogonal coordi-
nates mj , then these equations can be written as
ϕ(λ1,...,λn)(x) = ϕ(λn,...,λ1)(x), ϕ(λ1,...,λn)(x) = −ϕ(λn,...,λ1)(x),
respectively. According to (4.11) and (4.12), if
(m1,m2, . . . ,mn+1) = −(mn+1,mn, . . . ,m1) (4.13)
Antisymmetric Orbit Functions 23
(that is, the element λ has in the ω-basis the coordinates (λ1, λ2, . . . , λ2, λ1)), then the orbit
function ϕλ is real for n = 4k − 1, 4k and pure imaginary for n = 4k + 1, 4k + 2. In the first
case the antisymmetric orbit function can be represented as a sum of cosines of angles and in
the second case as a sum of sines of angles multiplied by i =
√
−1.
Proposition 2. In the orthogonal coordinates, antisymmetric orbit functions of An can be
represented as determinants of certain matrices:
ϕ(m1,m2,...,mn+1)(x) = det
(
e2πimixj
)n+1
i,j=1
≡ det
e2πim1x1 e2πim1x2 · · · e2πim1xn+1
e2πim2x1 e2πim2x2 · · · e2πim2xn+1
· · · · · · · · · · · ·
e2πimn+1x1 e2πimn+1x2 · · · e2πimn+1xn+1
. (4.14)
Proof. A proof of this formula follows from the fact that this expression for ϕ(m1,m2,...,mn+1)(x)
coincides with the expression given by the formula (4.9); see [30]. �
Taking into account the form of the half-sum of positive roots ρ for An, we can write down the
orbit function ϕρ(x), corresponding to the weight ρ = 1
2
∑
α>0
α, in the form of the Vandermonde
determinant,
ϕρ(x) = det
(
e2πiixj
)n+1
i,j=1
=
∏
k<l
(e2πixk − e−2πixl). (4.15)
The last equality follows from the expression for the Vandermonde determinant.
4.5 Antisymmetric orbit functions of Bn
Let λ = (m1,m2, . . . ,mn) be a strictly dominant element for Bn in orthogonal coordinates
described in Subsection 3.4. Then m1 > m2 > · · · > mn > 0. The Weyl group W (Bn) consists of
permutations of the coordinates mi with sign alternations of some of them. Representing points
x ∈ En also in the orthogonal coordinate system, x = (x1, x2, . . . , xn), and using formula (4.3)
we find that
ϕλ(x) =
∑
εi=±1
∑
w∈Sn
(detw)ε1ε2 . . . εne2πi〈w(ε1m1,...,εnmn),(x1,...,xn)〉
=
∑
εi=±1
∑
w∈Sn
(detw)ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn), (4.16)
where (w(ελ))1, . . . , (w(ελ))n are the orthogonal coordinates of the points w(ελ) if ελ = (ε1m1,
. . . , εnmn).
Since in W (Bn) there exists an element which changes signs of all coordinates mi, then for
each summand e2πi((w(ελ))1x1+···+(w(ελ))nxn) in the expressions (4.16) for the antisymmetric orbit
function ϕ(m1,m2,...,mn)(x) there exists exactly one summand complex conjugate to it, that is, the
summand e−2πi(((w(ελ))1x1+···+(w(ελ))nxn). This summand is with sign (−1)n = (detw′), where
w′ changes signs of all coordinates. Therefore, (detw′) = 1 if n = 2k and (detw′) = −1 if
n = 2k + 1. This means that antisymmetric orbit functions of Bn are real if n = 2k and pure
imaginary if n = 2k + 1. Each antisymmetric orbit function of Bn can be represented as a sum
of cosines of the corresponding angles if n = 2k and as a sum of sines, multiplied by i =
√
−1,
if n = 2k + 1. The following proposition is true [30]:
24 A. Klimyk and J. Patera
Proposition 3. Antisymmetric orbit functions of Bn can be represented in the form
ϕ(m1,m2,...,mn)(x) = det
(
e2πimixj − e−2πimixj
)n
i,j=1
= (2i)n det (sin 2πmixj)
n
i,j=1 . (4.17)
Proof. Let us take on the right hand side of (4.16) the sum of terms with fixed w ∈ Sn. It can
be written down as
(detw)
∑
εi=±1
ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn)
= (detw)(e2πi(wλ)1x1 − e−2πi(wλ)1x1) · · · (e2πi(wλ)nxn − e−2πi(wλ)nxn)
= (detw)(2i)n sin 2π(wλ)1x1 · . . . · sin 2π(wλ)nxn.
Then for ϕλ(x) we have
ϕλ(x) = (2i)n
∑
w∈Sn
(detw) sin 2π(wλ)1x1 · . . . · sin 2π(wλ)nxn
= (2i)n det (sin 2πmixj)
n
i,j=1 ,
where λ = (m1,m2, . . . ,mn). Proposition is proved. �
For the antisymmetric orbit function ϕρ, corresponding to the half-sum ρ of positive roots
of Bn, one has
ϕρ(x) = (2i)n det (sin 2πρixj)
n
i,j=1 ,
where ρ = (ρ1, ρ2, . . . , ρn) = (n− 1
2 , n−
3
2 , . . . ,
1
2).
4.6 Antisymmetric orbit functions of Cn
Let λ = (m1,m2, . . . ,mn) be a strictly dominant element for Cn in the orthogonal coordinates
described in Subsection 3.5. Then m1 > m2 > · · · > mn > 0. The Weyl group W (Cn) consists
of permutations of the coordinates with sign alternations of some of them. Representing points
x ∈ En also in the orthogonal coordinate system, x = (x1, x2, . . . , xn), we find that
ϕλ(x) =
∑
εi=±1
∑
w∈Sn
(detw)ε1ε2 . . . εne2πi〈w(ε1m1,...,εnmn),(x1,··· ,xn)〉
=
∑
εi=±1
∑
w∈Sn
(detw)ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn), (4.18)
where, as above, (w(ελ))1, . . . , (w(ελ))n are the orthogonal coordinates of the points w(ελ) if
ελ = (ε1m1, . . . , εnmn).
As in the case of Bn, in the expressions (4.18) for the functions ϕ(m1,m2,...,mn)(x) for each
summand e2πi((w(ελ))1x1+···+(w(ελ))nxn) there exists exactly one summand complex conjugate to
it, that is, the summand e−2πi((w(ελ))1x1+···+(w(ελ))nxn). Moreover, this summand is with sign
“+” if n = 2k and with sign “−” if n = 2k + 1. Therefore, antisymmetric orbit functions of Cn
are real if n = 2k and pure imaginary if n = 2k + 1.
Note that in the orthogonal coordinates the antisymmetric orbit functions ϕ(m1,m2,...,mn)(x)
of Cn coincides with the antisymmetric orbit functions ϕ(m1,m2,...,mn)(x) of Bn, that is, anti-
symmetric orbit functions (4.16) and (4.18) coincide. However, α-coordinates of the element
(m1,m2, . . . ,mn) for Cn do not coincide with α-coordinates of the element (m1,m2, . . . ,mn)
for Bn, that is, in α-coordinates the corresponding antisymmetric orbit functions of Bn and Cn
are different.
Antisymmetric Orbit Functions 25
Proposition 4. Antisymmetric orbit functions of Cn can be represented in the form
ϕ(m1,m2,...,mn)(x) = det
(
e2πimixj − e−2πimixj
)n
i,j=1
= (2i)n det (sin 2πmixj)
n
i,j=1 . (4.19)
Proof. This proposition follows from Proposition 3 if we take into account that antisymmetric
orbit functions of Cn and of Bn coincide in the orthogonal coordinate systems. �
For the antisymmetric orbit function ϕρ, corresponding to the half-sum of positive roots
of Cn, one has
ϕρ(x) = (2i)n det (sin 2πρixj)
n
i,j=1 ,
where ρ = (ρ1, ρ2, . . . , ρn) = (n, n− 1, . . . , 1).
4.7 Antisymmetric orbit functions of Dn
Let λ = (m1,m2, . . . ,mn) be a strictly dominant element for Dn in the orthogonal coordinates
described in Subsection 3.6. Then m1 > m2 > · · · > mn−1 > |mn|. The Weyl group W (Dn)
consists of permutations of the coordinates with sign alternations of even number of them.
Representing points x ∈ En also in the orthogonal coordinate system, x = (x1, x2, . . . , xn), and
using formula (4.3) we find that
ϕλ(x) =
∑
εi=±1
′ ∑
w∈Sn
(detw)e2πi〈w(ε1m1,...,εnmn),(x1,...,xn)〉
=
∑
εi=±1
′ ∑
w∈Sn
(detw)e2πi((w(ελ))1x1+···+(w(ελ))nxn), (4.20)
where (w(ελ))1, . . . , (w(ελ))n are the orthogonal coordinates of the points w(ελ) and the prime
at the sum sign means that the summation is over values of εi with even number of sign minus.
We have taken into account that an alternation of coordinates without any permutation does
not change a determinant.
Let mn 6= 0. Then in the expressions (4.20) for the orbit function ϕ(m1,m2,...,mn)(x) of Dn=2k
for each summand e2πi((w(ελ))1x1+···+(w(ελ))nxn) there exists exactly one summand (with the same
sign) complex conjugate to it. This means that these antisymmetric orbit functions of D2k are
real. Each orbit function of D2k can be represented as a sum of cosines of the corresponding
angles.
It is also proved by using the formula (4.20) that for mn 6= 0 the antisymmetric orbit functions
ϕ(m1,...,m2k,m2k+1)(x) and ϕ(m1,...,m2k,−m2k+1)(x) of D2k+1 are complex conjugate.
If mn = 0, then it follows from (4.20) that antisymmetric orbit functions of Dn are real and
can be represented as a sum of cosines of certain angles.
Explicit forms of antisymmetric orbit functions of Dn are described by the following propo-
sition [30]:
Proposition 5. Antisymmetric orbit functions of Dn are representable in the form
ϕ(m1,m2,...,mn)(x) = 1
2 det
(
e2πimixj−e−2πimixj
)n
i,j=1
+1
2 det
(
e2πimixj+e−2πimixj
)n
i,j=1
= 1
2(2i)n det (sin 2πmixj)
n
i,j=1 + 2n−1 det (cos 2πmixj)
n
i,j=1 (4.21)
if mn 6= 0, and in the form
ϕ(m1,m2,...,mn)(x) = 1
2 det
(
e2πimixj + e−2πimixj
)n
i,j=1
= 2n−1 det (cos 2πmixj)
n
i,j=1 (4.22)
if mn = 0.
26 A. Klimyk and J. Patera
Proof. Let mn 6= 0. We take on the right hand side of (4.20) a sum of terms with fixed w ∈ Sn.
It can be written as
Iw ≡ (detw)
∑
εi=±1
′
ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn),
where ε1ε2 · · · εn = 1 (since there is an even number of εi with εi = −1). A value of Iw does not
change if we add and subtract the same term to it:
Iw = (detw)
∑
εi=±1
′
ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn)
+ 1
2(detw)
∑
εi=±1
′′
ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn)
− 1
2(detw)
∑
εi=±1
′′
ε1ε2 · · · εne2πi((w(ελ))1x1+···+(w(ελ))nxn),
the sum with two primes here means that the summation is over values of εi with an odd number
of negative εi. We split down the right hand side into two parts,
Iw =
[
1
2(detw)
∑
εi=±1
′
· · · − 1
2(detw)
∑
εi=±1
′′
· · ·
]
+
[
1
2(detw)
∑
εi=±1
′
· · ·+ 1
2(detw)
∑
εi=±1
′′
· · ·
]
and repeat the reasoning of the proof of Proposition 3. As a result, we have
Iw = 1
2(detw)(e2πi(wλ)1x1 − e−2πi(wλ)1x1) · · · (e2πi(wλ)nxn − e−2πi(wλ)nxn)
+ 1
2(detw)(e2πi(wλ)1x1 + e−2πi(wλ)1x1) · · · (e2πi(wλ)nxn + e−2πi(wλ)nxn)
= 1
2(detw)(2i)n sin 2π(wλ)1x1 · · · · · sin 2π(wλ)nxn
+ 1
2(detw)2n cos 2π(wλ)1x1 · · · · · cos 2π(wλ)nxn.
Then for ϕλ(x) we have
ϕλ(x) = 1
2(2i)n
∑
w∈Sn
(detw) sin 2π(wλ)1x1 · · · · · sin 2π(wλ)nxn
+ 1
22n
∑
w∈Sn
(detw) cos 2π(wλ)1x1 · · · · · cos 2π(wλ)nxn
= 1
2(2i)n det (sin 2πmixj)
n
i,j=1 + 2n−1 det (cos 2πmixj)
n
i,j=1 ,
where λ = (m1,m2, . . . ,mn). Thus, the proposition is proved for the case mn 6= 0.
Let mn = 0. Then, in the expression (4.20) for ϕλ(x), in each term of the sum there exists
the multiplier e2πiεnmnxi with some i. Since mn = 0 we have e2πiεnmnxi = 1 for εn = 1 and for
εn = −1. The case εn = 1 gives an even number of negative εi in the set (ε1, ε2, . . . , εn−1) and
the case εn = −1 gives an odd number of negative εi in this set. Therefore, we may throw out
the sum over εn and consider that summation in (4.20) runs over εi = ±1, i = 1, 2, . . . n − 1
(with even or odd number of negative εi):
ϕλ(x) =
∑
εi=±1
∑
w∈Sn
(detw)e2πi((w(ελ))1x1+···+(w(ελ))nxn), (4.23)
where in the sum the multipliers, containing mn, are removed and the first sum does not contain
summation over εn. As in the case of the proof of Proposition 3, we take in (4.23) terms with
fixed w and write down it as
(detw)
∑
εi=±1
e2πi((w(ελ))1x1+···+(w(ελ))nxn)
Antisymmetric Orbit Functions 27
= 1
2(detw)(e2πi(wλ)1x1 + e−2πi(wλ)1x1) · · · (e2πi(wλ)nxn + e−2πi(wλ)nxn)
= 2n−1(detw) cos 2π(wλ)1x1 · · · · · cos 2π(wλ)nxn,
where, as before, the multipliers, containing mn, are omitted. Since e2πimnxi + e−2πimnxi = 2,
this leads to the formula (4.22) for ϕλ(x) when mn = 0. The proposition is proved. �
For the antisymmetric orbit function ϕρ, ρ = 1
2
∑
α>0
α, one has
ϕρ(x) = 2n−1 det (cos 2πρixj)
n
i,j=1 ,
where ρ = (ρ1, ρ2, . . . , ρn) = (n− 1, n− 2, . . . , 1, 0).
4.8 Symmetric orbit functions of Bn, Cn and Dn
In Subsections 4.5–4.7 we have derived expressions for antisymmetric orbit functions ϕλ(x)
of Bn, Cn and Dn in orthogonal coordinates as determinants of sine and cosine functions.
Similar expressions can be derived for symmetric orbit functions of Bn, Cn and Dn. Since in
the defining expressions for symmetric orbit functions φλ(x) (see formula (4.1)) there are no
multipliers (detw), then φλ(x) is expressed in terms of products of sine and cosine functions
(instead of determinants).
As before, we express elements λ = (m1,m2, . . . ,mn) and x = (x1, x2, . . . , xn) in the corre-
sponding orthogonal coordinate systems. In Propositions 6 and 7 below we suppose that λ is
an element of the dominant Weyl chamber and is not obligatory integral.
Proposition 6. Symmetric orbit functions of Bn and Cn can be represented in the form
φ(m1,m2,...,mn)(x) = 2n
∑
w∈Sn
cos 2πmw(1)x1 · · · · · cos 2πmw(n)xn, (4.24)
where w(1), w(2), . . . , w(n) is the set of numbers 1, 2, . . . , n obtained by acting by the permutation
w ∈ Sn.
This proposition is proved in the same way as Proposition 3 and we omit it.
Proposition 7. Symmetric orbit functions of Dn can be represented in the form
φ(m1,m2,...,mn)(x) = 2n−1
∑
w∈Sn
cos 2πmw(1)x1 · · · · · cos 2πmw(n)xn
+ 1
2(2i)n
∑
w∈Sn
sin 2πmw(1)x1 · · · · · sin 2πmw(n)xn (4.25)
if mn 6= 0 and in the form
ϕ(m1,m2,...,mn)(x) = 2n−1
∑
w∈Sn
cos 2πmw(1)x1 · · · · · cos 2πmw(n)xn (4.26)
if mn = 0, where w(1), w(2), . . . , w(n) is the set of numbers 1, 2, . . . , n obtained by acting by the
permutation s ∈ Sn.
A proof is similar to the proof of Proposition 5.
28 A. Klimyk and J. Patera
5 Properties of antisymmetric orbit functions
5.1 Anti-invariance with respect to the Weyl group
Since the scalar product 〈·, ·〉 in En is invariant with respect to the Weyl group W , that is,
〈wx,wy〉 = 〈x, y〉, w ∈W, x, y ∈ En,
antisymmetric orbit functions ϕλ for strictly positive elements λ (not obligatory belonging to P+
+ )
are anti-invariant with respect to W , that is,
ϕλ(w′x) = (detw′)ϕλ(x), w′ ∈W.
Indeed,
ϕλ(w′x) =
∑
w∈W
(detw)e2πi〈wλ,w′x〉 =
∑
w∈W
(detw)e2πi〈w′−1wλ,x〉
= (detw′)
∑
w∈W
(detw)e2πi〈wλ,x〉 = (detw′)ϕλ(x)
since w′−1w runs over the whole group W when w runs over W .
5.2 Anti-invariance with respect to the affine Weyl group
The affine Weyl group W aff is generated by reflections r0, rα1 , . . . , rαn (see Subsection 2.5). We
say that an antisymmetric orbit function ϕλ, λ ∈ P+
+ , is anti-invariant with respect to the affine
Weyl group W aff if ϕλ(r0x) = −ϕλ(x). Let us show that ϕλ(x) satisfies this relation. Since
r0x = rξx+ ξ∨, where ξ is the highest root (see Subsection 2.4), for µ ∈ P we have
〈µ, r0x〉 = 〈µ, ξ∨ + rξx〉 =
2〈µ, ξ〉
〈ξ, ξ〉
+ 〈µ, rξx〉 = integer + 〈rξµ, x〉
since r2ξ = 1. Hence,
ϕλ(r0x) =
∑
w∈W
(detw)e2πi〈wλ,r0x〉 =
∑
w∈W
(detw)e2πi〈rξwλ,x〉
= (det rξ)
∑
w∈W
(detw)e2πi〈wλ,x〉 = −ϕλ(x)
since rξ is a reflection belonging to W .
If λ 6= P , then ϕλ is not anti-invariant with respect to r0. It is anti-invariant only under
action by elements of the Weyl group W .
Due to the anti-invariance of antisymmetric orbit functions ϕλ, λ ∈ P+
+ , with respect to the
group W aff , it is enough to consider them only on the fundamental domain F ≡ F (W aff) of W aff .
Values of ϕλ on other points of En are determined by using the action of W aff on F or taking
a limit. In particular, functions ϕλ, λ ∈ P+
+ , are anti-invariant under a reflection with respect
to any (n− 1)-dimensional wall of the fundamental domain F .
5.3 Continuity and vanishing
An antisymmetric orbit function ϕλ is a finite sum of exponential functions. Therefore it is
continuous and has continuous derivatives of all orders in En.
Due to anti-invariance of the orbit functions ϕλ, λ ∈ D+
+, with respect to the Weyl group W ,
ϕλ vanishes on all walls of the Weyl chambers. The anti-invariance ϕλ(r0x) = −ϕλ(x) for
λ ∈ P+
+ shows that ϕλ, λ ∈ P+
+ , vanishes on the boundary of the fundamental domain F of the
affine Weyl group W aff .
Antisymmetric Orbit Functions 29
5.4 Realness and complex conjugation
The results, formulated below and concerning antisymmetric orbit functions of the Coxeter–
Dynkin diagrams An, Bn, Cn, Dn and G2, were proved in the previous section.
Antisymmetric orbit functions of the following Coxeter–Dynkin diagrams are real:
B2k, C2k, G2.
Antisymmetric orbit functions of the Coxeter–Dynkin diagrams B2k+1 and C2k+1 are purely
imaginary. The antisymmetric orbit functions ϕλ of the Coxeter–Dynkin diagrams A4k−1 and
A4k satisfy the condition
ϕ(λ1,λ2,...,λn)(x) = ϕ(λn,λn−1,...,λ1)(x)
and antisymmetric orbit functions ϕλ of the Coxeter–Dynkin diagrams A4k+1 and A4k+2 satisfy
the condition
ϕ(λ1,λ2,...,λn)(x) = −ϕ(λn,λn−1,...,λ1)(x).
Antisymmetric orbit functions ϕ(λ1,λ2,...,λn) of Dn are real if λn−1 = λn. If λn−1 6= λn, then
orbit functions ϕ(λ1,λ2,...,λn) are real for n = 2k and satisfy the condition
ϕ(λ1,...,λn−2,λn−1,λn) = ϕ(λ1,...,λn−2,λn,λn−1)
if n = 2k + 1.
5.5 Scaling symmetry
Let O(λ) be an orbit of λ, λ ∈ D+
+. Since w(cλ) = cw(λ) for any c ∈ R and for any w ∈ W ,
then the orbit O(cλ) is an orbit consisting of the points cwλ, w ∈ W . Moreover, points wλ
and cwλ of the signed orbits O±(λ) and O±(cλ), respectively, have the same sign. Let ϕλ =∑
w∈W
(detw)e2πiwλ be the antisymmetric orbit function for λ ∈ D+
+. Then
ϕcλ(x) =
∑
w∈W
(detw)e2πi〈cwλ,x〉 =
∑
w∈W
(detw)e2πi〈wλ,cx〉 = ϕλ(cx).
The equality ϕcλ(x) = ϕλ(cx) expresses the scaling symmetry of orbit functions.
If we consider only antisymmetric orbit functions ϕλ, corresponding to λ ∈ P+
+ , then the
scaling symmetry ϕcλ(x) = ϕλ(cx) holds for those values c ∈ R\{0} for which cλ ∈ P+
+ .
5.6 Duality
Due to the invariance of the scalar product 〈·, ·〉 with respect to the Weyl group W , 〈wµ,wy〉 =
〈µ, y〉, for x ∈ En not lying on a wall of some Weyl chamber we have
ϕλ(x) =
∑
w∈W
(detw)e2πi〈λ,w−1x〉 =
∑
w∈W
(detw)e2πi〈λ,wx〉 = ϕx(λ).
The relation ϕλ(x) = ϕx(λ) expresses the duality of antisymmetric orbit functions.
30 A. Klimyk and J. Patera
5.7 Orthogonality
Antisymmetric orbit functions ϕλ, λ ∈ P+
+ , are orthogonal on F with respect to the Euclidean
measure [31]:
|F |−1
∫
F
ϕλ(x)ϕλ′(x)dx = |W |δλλ′ , (5.1)
where the overbar means complex conjugation. This relation directly follows from orthogonality
of the exponential functions e2πi〈µ,x〉 (entering into the definition of orbit functions) for different
weights µ and from the fact that a given weight ν ∈ P belongs to precisely one orbit function.
In (5.1), |F | means an area of the fundamental domain F .
Sometimes, it is difficult to find the area |F |. In this case it is useful the following form of
the formula (5.1):∫
T
ϕλ(x)ϕλ′(x)dx = |W |δλλ′ ,
where T is the torus in En described in Subsection 9.1 below. If to assume that an area of T is
equal to 1, |T| = 1, then |F | = |W |−1 and formula (5.1) takes the form∫
F
ϕλ(x)ϕλ′(x)dx = δλλ′ . (5.2)
The formula (5.2) gives the orthogonality relation for the antisymmetric multivariate sine
function (4.19). We have
22n
∫
F
det (sin 2πmixj)
n
i,j=1 det
(
sin 2πm′
ixj
)n
i,j=1
dx = δm,m′ , (5.3)
where m = (m1,m2, . . . ,mn) and m′ = (m′
1,m
′
2, . . . ,m
′
n) are strictly dominant and integral
(that is, mi,m
′
j ∈ Z>0, m1 > m2 > · · · > mn > 0), and the domain F consists of points
x = (x1, x2, . . . , xn) ∈ En such that
1
2
≥ x1 ≥ x2 ≥ · · · ≥ xn ≥ 0
(see Subsection 5.10 below).
A similar orthogonality relation can be written down for the symmetric multivariate cosine
function (4.24). Introducing the notation
det+(cos 2πmixj)ni,j=1 :=
∑
w∈Sn
cos 2πmw(1)x1 · · · · · cos 2πmw(n)xn (5.4)
we have
22n
∫
F
det+ (cos 2πmixj)
n
i,j=1 det+
(
cos 2πm′
ixj
)n
i,j=1
dx = |Wm|δm,m′ , (5.5)
where m and m′ are such that m1 ≥ m2 ≥ · · · ≥ mn, m′
1 ≥ m′
2 ≥ · · · ≥ m′
n, and |Wm| is an
order of the subgroup |Wm| ⊂ Sn consisting of elements leaving m invariant.
Antisymmetric Orbit Functions 31
5.8 Orthogonality to symmetric orbit functions
Let αi be a simple root. We create the domain F ext = F ∪ rαiF , where rαi is the reflection
corresponding to the root αi and F is the fundamental domain. Since for µ ∈ P+
+ we have
φµ(rαix) = φµ(x), ϕµ(rαix) = −ϕµ(x),
where φµ is a symmetric orbit function, then∫
F ext
φµ(x)ϕµ(x)dx = 0. (5.6)
Indeed, due to symmetry and antisymmetry of symmetric and antisymmetric orbit functions we
have ∫
F ext
φµ(x)ϕµ(x)dx =
∫
F
φµ(x)ϕµ(x)dx+
∫
rαiF
φµ(x)ϕµ(x)dx
=
∫
F
φµ(x)ϕµ(x)dx+
∫
F
φµ(x)(−ϕµ(x))dx = 0.
For the case of A1 the orthogonality (5.6) means the orthogonality of the functions sine and
cosine on the interval (0, 2π).
The formula (5.6) determines orthogonality of multivariate sine and cosine functions (4.19)
and (5.4):∫
F ext
det (sin 2πmixj)
n
i,j=1 det+(cos 2πmixj)ni,j=1dx = 0,
where the notations are such as in (5.3) and F ext consists of points x ∈ En such that
1
2 ≥ x1 ≥ x2 ≥ · · · ≥ xn ≥ 0 or 1
2 ≥ x2 ≥ x1 ≥ x3 ≥ x4 ≥ · · · ≥ xn ≥ 0.
5.9 Antisymmetric orbit functions ϕρ
Let ρ be the half-sum of all positive roots of a root system:
ρ =
1
2
∑
α>0
α. (5.7)
It is well-known that for all simple Lie algebras in ω-coordinate we have
ρ = ω1 + ω2 + · · ·+ ωn ≡ (1 1 · · · 1).
The antisymmetric orbit function ϕρ is important in the theory of characters of group represen-
tations. We have
ϕρ(x) =
∑
w∈W
(detw)e2πi〈wρ,x〉. (5.8)
Proposition 8. The antisymmetric orbit function ϕρ can be represented as
ϕρ(x) =
∏
α>0
(eπi〈α,x〉 − e−πi〈α,x〉) = e2πi〈ρ,x〉
∏
α>0
(1− e−2πi〈α,x〉)
= (2i)r
∏
α>0
sinπ〈α, x〉, (5.9)
where r is a number of positive roots in the corresponding root system.
32 A. Klimyk and J. Patera
Proof. The expression g(x) =
∏
α>0
(eπi〈α,x〉 − e−πi〈α,x〉) is antisymmetric with respect to the
Weyl group W . Indeed, if ri is a reflection, corresponding to the simple root αi, then due to
Proposition 1 we have
g(rix) =
∏
α>0
(eπi〈α,rix〉 − e−πi〈α,rix〉) =
∏
α>0
(eπi〈riα,x〉 − e−πi〈riα,x〉)
= (e−πi〈αi,x〉 − eπi〈αi,x〉)
∏
α>0,α 6=αi
(eπi〈α,x〉 − e−πi〈α,x〉) = −g(x).
Since αi is an arbitrary simple root, g(x) is antisymmetric with respect to W . Then g(x) is
a sum of antisymmetric orbit functions. Representing
∏
α>0(e
πi〈α,x〉−e−πi〈α,x〉) in the form (4.3),
we see that in this form there exists only one term eπi〈λ,x〉 with a strictly dominant weight λ
and this weight coincides with ρ. Therefore, g(x) = ϕρ(x). Proposition is proved. �
An invariant measure of the compact Lie group G, associated with the Weyl group W ,
is expressed in terms of |ϕρ|2. Taking into account an explicit form of positive roots in the
orthogonal coordinate systems, it is easy to derive from (5.9) that in these coordinates we have
ϕρ(x) = (2i)n(n+1)/2
∏
1≤i<j≤n+1
sinπ(xi − xj) for An,
ϕρ(x) = (2i)n
2
∏
1≤i<j≤n
sinπ(xi − xj) sinπ(xi + xj)
∏
1≤i≤n
sinπxi for Bn,
ϕρ(x) = (2i)n
2
∏
1≤i<j≤n
sinπ(xi − xj) sinπ(xi + xj)
∏
1≤i≤n
sin 2πxi for Cn,
ϕρ(x) = (2i)n(n−1)
∏
1≤i<j≤n
sinπ(xi − xj) sinπ(xi + xj) for Dn.
These formulas give other expressions for |ϕρ(x)| with respect to formulas derived in Subsec-
tions 4.4–4.7.
Proposition 9. The antisymmetric orbit function ϕρ vanishes on the boundary of the funda-
mental domain F . It does not vanish on intrinsic points of F .
Proof. Since ρ ∈ P+
+ , then ϕρ vanishes on the boundary of F due to the results of Subsec-
tion 5.3. From the other side, it is easy to see from (5.9) that the set of points, on which ϕρ(x)
vanishes, coincides with the set of all points x ∈ En for which 〈α, x〉 ∈ Z for some root α. No of
these points is an intrinsic point of F . The proposition is proved. �
From Proposition 9 and from the above formulas for ϕρ(x) we easily derive explicit forms of
the fundamental domains for the cases An, Bn, Cn, Dn in the orthogonal coordinates.
(a) The fundamental domain F (An) is contained in the domain of real points x = (x1, x2, . . . ,
xn+1) such that
x1 > x2 > · · · > xn+1, x1 + x2 + · · ·+ xn+1 = 0.
Moreover, a point x of this domain belongs to F (An) if and only if x1 + |xn+1| < 1. The last
condition means in fact the relation 〈x, ξ〉 < 1 for the highest root ξ of the root system An.
(b) The fundamental domain F (Bn) is contained in the domain of points x = (x1, x2, . . . ,
xn) such that
1 > x1 > x2 > · · · > xn > 0.
Antisymmetric Orbit Functions 33
Moreover, a point x of this domain belongs to F (Bn) if and only if x1 + x2 < 1.
(c) The fundamental domain F (Cn) consists of all points x = (x1, x2, . . . , xn) such that
1
2 > x1 > x2 > · · · > xn > 0.
(d) The fundamental domain F (Dn) is contained in the domain of points x = (x1, x2, . . . ,
xn) such that
1 > x1 > x2 > · · · > xn−1 > |xn|.
Moreover, a point x of this domain belongs to F (Dn) if and only if x1 + x2 < 1.
5.10 Symmetric orbit functions φρ
Let us consider a symmetric counterpart of the antisymmetric orbit function ϕρ(x):
φρ(x) :=
∑
w∈W
e2πi〈wρ,x〉, (5.10)
where, as before, ρ = 1
2
∑
α>0
α.
Proposition 10. The symmetric orbit function φρ can be represented as
ϕρ(x) =
∏
α>0
(eπi〈α,x〉 + e−πi〈α,x〉) = e2πi〈ρ,x〉
∏
α>0
(1 + e−2πi〈α,x〉)
= 2r
∏
α>0
cosπ〈α, x〉, (5.11)
where r is a number of positive roots in the corresponding root system.
This proposition is proved in the same way as Proposition 6.
It is easy to derive from (5.11) that in the orthogonal coordinate systems we have
φρ(x) = 2n(n+1)/2
∏
1≤i<j≤n+1
cosπ(xi − xj) for An,
φρ(x) = 2n
2
∏
1≤i<j≤n
cosπ(xi − xj) cosπ(xi + xj)
∏
1≤i≤n
cosπxi for Bn,
φρ(x) = 2n
2
∏
1≤i<j≤n
cosπ(xi − xj) cosπ(xi + xj)
∏
1≤i≤n
cos 2πxi for Cn,
φρ(x) = 2n(n−1)
∏
1≤i<j≤n
cosπ(xi − xj) cosπ(xi + xj) for Dn.
6 Properties of antisymmetric orbit functions of An
By using results on decomposition of certain reducible representations of the group GL(n,C)
into irreducible constituents, properties of antisymmetric orbit functions of An can be derived.
We use in this section the results of [30].
34 A. Klimyk and J. Patera
6.1 Decomposition of symmetric powers of representations
We need some formulas for decomposition of symmetric powers of finite dimensional irreducible
representations of the group GL(n1,C) × GL(n2,C). Let us first define symmetric powers of
representations.
Let T be a finite dimensional representation of a group G on a linear space X. It induces
a representation on the space Pm(X), which is a subspace of the space P(X) of polynomials onX,
consisting of all homogeneous polynomials of power m. In order to determine this representation
we note that the formula
(Q(g)p)(x) = p(T (g−1)x), x ∈ X, g ∈ G,
gives a representation of G on P(X) which is denoted by Q. The subspace Pm(X) is invariant
with respect to the representation Q. The restriction of Q onto Pm(X) is called an m-th
symmetric power of the representation T and is denoted as σm(T ).
Let G = GL(n1,C)×GL(n2,C), n1 ≤ n2. Then a finite dimensional irreducible representa-
tion of G is given (in the orthogonal coordinate system) by
(λ|µ) = (m1,m2, . . . ,mn1 | r1, r2, . . . , rn2),
where
m1 ≥ m2 ≥ · · · ≥ mn1 , r1 ≥ r2 ≥ · · · ≥ rn2 .
We assume that mn1 ≥ 0 and rn2 ≥ 0.
One can consider three representations of G = GL(n1,C)×GL(n2,C) on the space Mn1n2(C)
of n1 × n2 complex matrices. They are given by the formulas
T (g1, g2)X = g1Xg
t
2, (6.1)
T ′(g1, g2)X = g1Xg
∗
2, (6.2)
T ′′(g1, g2)X = g1Xg
−1
2 , (6.3)
where the index t means a transposition and ∗ means a transposition together with complex
conjugation. Then for symmetric powers of these representations we have the following decom-
positions into irreducible representations of G (a proof see in [30]):
σm(T ) =
∑
∑
i si=m
(s1, s2, . . . , sn1 | s1, s2, . . . , sn1 , 0, . . . , 0), (6.4)
σm(T ′) =
∑
∑
i si=m
(s1, s2, . . . , sn1 | s1, s2, . . . , sn1 , 0, · · · , 0), (6.5)
σm(T ′′) =
∑
∑
i si=m
(s1, s2, . . . , sn1 | 0, . . . , 0,−sn1 ,−sn2 , . . . ,−s1), (6.6)
where summations are over all s1, s2, . . . , sn1 such that
s1 ≥ s2 ≥ · · · ≥ sn1 ≥ 0, s1 + s2 + · · ·+ sn1 = m
and the overbar means that a representation is anti-analytic.
If n1 = n2 = n, then the formulas (6.1)–(6.3) determine the following tensor product repre-
sentations of G = GL(n,C) on the space Mn(C) of complex n× n matrices:
(T1 ⊗ T1)(g)X = gXgt,
Antisymmetric Orbit Functions 35
(T1 ⊗ T̄1)(g)X = gXg∗,
(T1 ⊗ T̂1)(g)X = gXg−1,
where T1 is the first fundamental representation (with highest weight (1, 0, . . . , 0)) of G =
GL(n,C), T̄1 and T̂1 are the complex conjugate and contragredient representations to the rep-
resentation T1, respectively. Then according to (6.4)–(6.6) we have
σm(T1 ⊗ T1) =
∑
m
Tm ⊗ Tm, (6.7)
σm(T1 ⊗ T̄1) =
∑
m
Tm ⊗ T̄m, (6.8)
σm(T1 ⊗ T1) =
∑
m
Tm ⊗ T̂m, (6.9)
where Tm is the irreducible representation of the group GL(n,C) with highest weight m ≡
(m1,m2, . . . ,mn), m1 ≥ m2 ≥ · · · ≥ mn ≥ 0, and the summation is over those m for which
m1 +m2 + · · ·+mn = m.
Replacing the space Mn(C) by the subspaces of all symmetric or all antisymmetric matrices
from Mn(C) we obtain the following decompositions of symmetric powers of the irreducible
representations of GL(n,C) with highest weights (2, 0, . . . , 0) and (1, 1, 0, . . . , 0), respectively:
σm(T(2,0,...,0)) =
∑
m
T(2m1,2m2,...,2mn), (6.10)
σm(T(1,1,0,...,0)) =
∑
m
T(m1,m1,...,mk,mk), (6.11)
where the summations are over those m = (m1,m2, . . . ,mn) for which m1 ≥ m2 ≥ · · · ≥ mn ≥ 0
and m1 +m2 + · · · +mn = m. In the second formula n = 2k; if n = 2k + 1, then on the right
hand side we have to replace T(m1,m1,...,mk,mk) by T(m1,m1,...,mk,mk,0).
6.2 Properties of antisymmetric orbit functions of An
We represent antisymmetric orbit functions of An in the orthogonal coordinate system as in
formula (4.9). Let λ = (m1,m2, . . . ,mn+1) and λ+ r = (m1 + r,m2 + r, . . . ,mn+1 + r), where r
is a fixed real number. If x = (x1, x2, . . . , xn+1), x1 + x2 + · · ·+ xn+1 = 0, and w ∈W , then we
have
e2πi〈λ+r,wx〉 = e2πi〈λ,wx〉e2πi〈0+r,wx〉 = e2πi〈λ,wx〉.
It follows from this equality that
ϕλ(x) = ϕλ+r(x), φλ(x) = φλ+r(x) (6.12)
where λ = (m1,m2, . . . ,mn+1) is given in the orthogonal coordinate system. This means that
instead of mi, i = 1, 2, . . . , n+1, determined by formulas of Subsection 3.3, we may assume that
m1,m2, . . . ,mn+1 are integers such that m1 ≥ m2 ≥ · · · ≥ mn+1 ≥ 0. We adopt this assumption
in this subsection.
For simplicity we introduce the following notations:
e2πixj = yj , j = 1, 2, . . . , n+ 1.
36 A. Klimyk and J. Patera
In order to receive relations for antisymmetric orbit functions we have to take characters of
representations on both sides of relations of the previous subsection and to substitute the ex-
pression
χλ(x) =
ϕλ+ρ(x)
ϕρ(x)
for characters (see Subsection 9.1 below).
The relation (6.4) gives the equality
χσm(T )(g1, g2) =
∑
∑
i mi=m
χs(g1)χs′(g2)
for characters of representations, where s′ = (s1, s2, . . . , sn1 , 0, . . . , 0) and the sum is such as
in (6.4). Multiply both sides by tm and sum up over all non-negative integral values of m. We
get
∞∑
m=0
tmχσm(T )(g1, g2) =
∞∑
m=0
tm
∑
∑
i
mi=m
χs(g1)χs′(g2). (6.13)
Take this relation for g1 = diag(y1, y2, . . . , yn1) and g2 = diag(z1, z2, . . . , zn2). For such g1 and g2
the left hand side takes the form
∞∑
m=0
tmχσm(T )(g1, g2) =
∞∑
m=0
tm
∑
yr11 y
r2
2 · · · yrn1
n1 z
p1
1 z
p2
2 · · · zpn2
n2 ,
where the second summation on the right hand side is over integral ri and pj such that
n1∑
i=1
ri +
n2∑
j=1
pj = m. Therefore, the relation (6.13) can be written as
∞∑
m=0
tm
∑
∑
i mi=m
χs(g1)χs′(g2) =
n1∏
i=1
n2∏
j=1
(1− tyizj)−1,
where g1 = diag(y1, y2, . . . , yn1), g2 = diag(z1, z2, . . . , zn2) and |t| < 1, |yi| < 1, |zj | < 1.
Further, we represent characters in (6.13) in terms of antisymmetric orbit functions and set
t = 1. As a result, we receive the following relation for antisymmetric orbit functions:
ϕ(n1)
ρ1 (y)ϕ(n2)
ρ2 (z)
n1∏
i=1
n2∏
j=1
(1− yizj)−1 =
∑
m
ϕ
(n1)
s+ρ1(y)ϕ
(n2)
ŝ+ρ2
(z), (6.14)
where y = (y1, y2, . . . , yn1), z = (z1, z2, . . . , zn2), and ρ1 and ρ2 are half-sums of positive roots
for An1−1 and An2−1, respectively. In particular, if n1 = n2 = n, then
ϕρ(y)ϕρ(z)
n∏
i,j=1
(1− yizj)−1 =
∑
m
ϕs+ρ(y)ϕs+ρ(z).
Applying to this relation the Cauchy lemma, which states that
det
(
(1− yizj)−1
)n
i,j=1
= ϕρ(y)ϕρ(z)
n∏
i,j=1
(1− yizj)−1,
Antisymmetric Orbit Functions 37
we obtain the identity∑
λ
ϕλ+ρ(y)ϕλ+ρ(z) = det
(
(1− yizj)−1
)n
i,j=1
.
Now we use the relation (6.10) for representations of the group GL(n,C) in order to derive
the equality
χ(2,0,...,0),m(g) =
∑
|m|=m
χ2m(g), g ∈ GL(n,C),
where on the left hand side is the character of the representation σm(T(2,0,...,0)) from (6.10),
2m = (2m1, 2m2, . . . , 2mn), m1 ≥ m2 ≥ · · · ≥ mn ≥ 0, |m| = m1 +m2 + · · · +mn. We derive
from this equality that
∞∑
m=0
χ(2,0,...,0),m(g)tm =
∞∑
m=0
tm
∑
|m|=m
χ2m(g). (6.15)
Setting g = diag(y1, y2, . . . , yn), for the left hand side we have the expression
∞∑
m=0
χ(2,0,...,0),m(g)tm =
∏
1≤i≤j≤n
(1− tyiyj)−1,
where |yi| < 1 and |t| < 1. Now using the Weyl character formula one receives∑
m
ϕ2m+ρ(y)t|m| = ϕρ(y)
∏
1≤i≤j≤n
(1− tyiyj)−1. (6.16)
Substituting into (6.16) the expression for ϕ2m+ρ(y), setting t = 1 and using the evident
relation∑
m
am1
1 am2
2 · · · amn
n = (1− a1)−1(1− a1a2)−1 · · · (1− a1a2 · · · an)−1,
we get a non-trivial identity
∑
(i1,i2,...,in)
sign(i1, i2, . . . , in) yn−1
i1
yn−2
i2
· · · yin−1
(1− y2
i1
)(1− y2
i1
y2
i2
) · · · (1− y2
i1
· · · y2
in
)
= ϕρ(y)
∏
1≤i≤j≤n
(1− yiyj)−1,
where summation is over all permutations (i1, i2, . . . , in) of the natural numbers 1, 2, . . . , n and
sign (i1, i2, . . . , in) means a sign of the permutation (i1, i2, . . . , in).
In the same way, from the decomposition (6.11) we derive∑
m
ϕm̃+ρ(y)t|m| = ϕρ(y)
∏
1≤i<j≤n
(1− tyiyj)−1, (6.17)
where summation is over m = (m1,m2, . . . ,mk), k = [n/2], m1 ≥ m2 ≥ · · · ≥ mk ≥ 0,
m̃ = (m1,m1,m2,m2, . . . ,mk,mk) (here one has to add 0 at the end if n = 2k + 1). As above,
we receive from here the identity
∑
(i1,i2,...,in)
sign(i1, i2, . . . , in) yn−1
i1
yn−2
i2
· · · yin−1
(1− yi1yi2)(1− yi1yi2yi3yi4) · · · (1− yi1 · · · yi2ν )
= ϕρ(y)
∏
1≤i≤j≤n
(1− yiyj)−1.
38 A. Klimyk and J. Patera
In order to derive other identities we note that for θ = (θ1, θ2, . . . , θn) one has
lim
θ→0
ϕλ+ρ(e2πiθ) = ϕρ(λ+ ρ).
Taking into account this relation and substituting y = (1, . . . , 1) and z = (1, . . . , 1) into (6.14),
we derive a relation for the antisymmetric orbit functions ϕρ:∑
|m|=m
ϕ(s)
ρs
(m + ρs)ϕ(r)
ρr
(m̂ + ρr) =
(sr +m− 1)!
(sr − 1)!m!
ϕ(s)
ρs
(ρs)ϕ(r)
ρr
(ρr), (6.18)
where r ≥ s ≥ 0, m = (m1,m2, . . . ,ms), m̂ = (m1,m2, . . . ,ms, 0, . . . , 0), |m| = m1 +m2 + · · ·+
ms and ρk = (k − 1, k − 2, . . . , 1, 0).
From the relations (6.16) and (6.17) one similarly obtains the identities
∑
|m|
ϕρ(2m + ρ) =
(m+ 1
2n(n+ 1)− 1)!
(1
2n(n+ 1)− 1)!m!
ϕρ(ρ), (6.19)
∑
|m|
ϕρ(m̃ + ρ) =
(m+ 1
2n(n− 1)− 1)!
(1
2n(n− 1)− 1)!m!
ϕρ(ρ), (6.20)
where m = (m1,m2, . . . ,mν), m̃ = (m1,m1,m2,m2, . . . ,mν ,mν), ν = [n/2].
From (6.18)–(6.20) we derive the relations∑
m
ϕ(s)
ρs
(m + ρs)ϕ(r)
ρr
(m̂ + ρr)t|m| = (1− t)−srϕ(s)
ρs
(ρs)ϕ(r)
ρr
(ρr),∑
m
ϕρ(2m + ρ)t|m| = (1− t)−n(n+1)/2ϕρ(ρ),∑
m
ϕρ(2m̃ + ρ)t|m| = (1− t)−n(n−1)/2ϕρ(ρ).
Other decompositions of the previous subsection lead to new relations for antisymmetric orbit
functions of An.
7 Decomposition of products of (anti)symmetric orbit functions
The aim of this section is to derive how to decompose products of (anti)symmetric orbit functions
into sums of (anti)symmetric orbit functions. Such operations are fulfilled by means of the
corresponding decompositions of (signed) orbits.
7.1 Products of symmetric and antisymmetric orbit functions
Invariance of symmetric orbit functions φλ and anti-invariance of antisymmetric orbit func-
tions ϕλ with respect to the Weyl group W lead to the following statements:
Proposition 11. (a) A product of symmetric orbit functions expands into a sum of symmetric
orbit functions:
φλφµ =
∑
ν
nνφν , (7.1)
where an integer nν shows how many times the orbit function φν is contained in the product φλφµ.
Antisymmetric Orbit Functions 39
(b) A product of antisymmetric orbit functions expands into a sum of symmetric orbit func-
tions:
ϕλϕµ =
∑
ν
nνφν , (7.2)
where nν are positive or negative integral coefficients.
(c) A product of symmetric and antisymmetric orbit functions expands into a sum of anti-
symmetric orbit functions:
φλϕµ =
∑
ν
nνϕν . (7.3)
where nν are positive or negative integral coefficients.
Proof. In the case (b) we have
ϕλ(wx)ϕµ(wx) = (detw)2ϕλ(x)ϕµ(x) = ϕλ(x)ϕµ(x).
Therefore, the product ϕλϕµ is invariant with respect to W . Hence, it can be expanded into
symmetric orbit functions (see Subsection 7.8 in [1]). Since antisymmetric orbit functions ϕλ
are sums of exponential functions e2πi〈σ,x〉 with coefficients ±1, the coefficients nν in (7.2) are
integers. In the case (c) we have a similar situation. The case (a) is considered in [1]. �
In order to fulfill expansions (a)–(c) in an explicit form it is necessary to fulfill the corre-
sponding decompositions of products of (signed) orbit, considering usual (not signed) orbits as
signed orbits in which to all points the sign “+” is assigned, and to take into account that
multiplication of (singed) orbit functions are reduced to multiplication of exponential functions.
A product O±(λ)⊗O±(λ′) of two signed orbits O±(λ) and O±(λ′) (one or two of them can
be replaced by usual orbits, that is, to the corresponding points the sign “+” is assigned) is the
set of all points of the form λ1 + λ2 (where λ1 ∈ O±(λ) and λ2 ∈ O±(λ′)) with a sign which is a
product of signs of λ and λ′. Since a set of points λ1+λ2 (without signs), λ1 ∈ O(λ), λ2 ∈ O(λ′),
is invariant with respect to action of the corresponding Weyl group, each product of orbits is
decomposable into a sum of orbits. Then it follows from assertions (a)–(c) of Proposition 11
that, considering points λ1 + λ2 with signs, we obtain decomposition of O±(λ) ⊗ O±(λ′) into
usual orbits, where to each point a sign is assigned (the same sign for points of a fixed orbit).
Moreover, a product of a signed orbit with a usual orbit decomposes into signed orbits (not into
usual orbits).
Under product of (signed) orbits we may receive an orbit with signed points in which the
sign “−” corresponds to a dominant weight. This means that we obtain a signed orbit with
opposite signs for its points. In this case we say that a product of (signed) orbits contains
the corresponding signed orbit with sign “−” and denote it by −O±(µ). That is why in (7.2)
and (7.3) negative coefficients can appear.
Example. Orbits of A1. If a ∈ E1 is strictly positive, then the signed orbit of this point O±(a)
consists of two signed points a+ and −a−. It is easy to see that for the product O±(a)⊗O±(b)
we have
O±(a)⊗O±(b) ≡ {a+,−a−} ⊗ {b+,−b−}
= {(a+ b)+, (−a− b)+} ∪ {(|a− b|)−, (−|a− b|)−}
= O(a+ b) ∪ −O(|a− b|),
where O(a+ b) and O(|a− b|) are usual (not signed) orbits. Therefore,
ϕa(x)ϕb(x) = φa+b(x)− φ|a−b|(x).
40 A. Klimyk and J. Patera
Similarly, we have
O±(a)⊗O(b) = O±(a+ b) ∪O±(a− b), a > b > 0,
O±(a)⊗O(b) = O±(a+ b) ∪ −O±(b− a), b > a > 0.
Thus,
ϕa(x)ϕb(x) = ϕa+b(x) + ϕa−b(x), a > b > 0,
ϕa(x)φb(x) = ϕa+b(x)− ϕ|a−b|(x), b > a > 0.
For the corresponding decompositions for O(a)⊗O(b) see [1].
Decomposition of products of orbits in higher dimension of the Euclidean space is not a simple
task. In the next subsection we consider some general results on the decomposition.
7.2 Products of symmetric and antisymmetric orbits
Let O(λ) = {wλ|w ∈ W/Wλ} be a usual orbit and O±(µ) = {wµ|w ∈ W} be a signed orbit.
Then
O(λ)⊗O±(µ) = {(wλ+ w′µ)detw′ |w ∈W/Wλ, w
′ ∈W}
= {(wλ+w1µ)detw1 |w∈W/Wλ} ∪ {(wλ+w2µ)detw2 |w∈W/Wλ} ∪ · · ·
∪ {(wλ+wsµ)detws |w∈W/Wλ}, (7.4)
where w1, w2, . . . , ws is the set of elements of W . Since a product of an orbit and a signed orbit
decomposes into signed orbits, for decomposition of the product O(λ) ⊗ O±(µ) into separate
signed orbits it is necessary to take dominant elements from each term of the right hand side
of (7.4). That is, O(λ)⊗O±(µ) is a union of the signed orbits, corresponding to points from
D({(wλ+ w1µ)detw1 |w ∈W/Wλ}), D({(wλ+ w2µ)detw2 |w ∈W/Wλ}), . . . ,
D({(wλ+ wsµ)detws |w ∈W/Wλ}), (7.5)
where D({(wλ+wiµ)detwi |w ∈W/Wλ}) means the set of dominant signed elements in {(wλ+
wiµ)detwi |w ∈W/Wλ}.
Proposition 12. The product O(λ)⊗O±(µ) consists only of signed orbits of the form O±(|wλ+
µ|), w ∈W/Wλ, where |wλ+µ| is a dominant weight of the orbit containing wλ+µ. Moreover,
each such orbit O±(|wλ+ µ|), w ∈W/Wλ, except for those of them, for which |wλ+ µ| lies on
some wall of the dominant Weyl chamber, belongs to the product O(λ)⊗O±(µ).
Proof. For each dominant element wλ+ wiµ from (7.5) there exists an element w′′ ∈ W such
that w′′(wλ+wiµ) = w′λ+ µ. It means that wλ+wiµ is of the form |w′λ+ µ|, w′ ∈W/Wλ. It
is clear that a sign of this dominant element is detwiw′′. Conversely, take any element wλ+ µ,
w ∈ W/Wλ. It belongs (with some sign) to the product O(λ) ⊗ O±(µ). This means that
|wλ+ µ| also belongs to this product. Therefore, the signed orbit O±(|wλ+ µ|) is contained in
O(λ)⊗O±(µ) with some coefficient if |wλ+ µ| does not lie on some wall of the dominant Weyl
chamber. This coefficient cannot vanish since if it vanishes, then in the product O(λ)⊗O±(µ)
there are contained points of O±(|wλ+µ|), but taken with an opposite signs. In this case there
exists another element w′λ+ µ such that wλ+ µ = w′′(w′λ+ µ). Since µ does not lie on a wall,
this is not possible. Proposition is proved. �
Antisymmetric Orbit Functions 41
It follows from Proposition 12 that for decomposition of the product O(λ) ⊗ O±(µ) into
separate signed orbits we have to take all elements wλ + µ, w ∈ W/Wλ, and to find the
corresponding strictly dominant elements |wλ+ µ|, w ∈W/Wλ.
Corollary. For the product O(λ)⊗O±(µ) we have
O(λ)⊗O±(µ) =
⋃′
w∈W/Wλ
εwO
±(|wλ+ µ|), (7.6)
where the prime means that the term with |wλ + µ| lying on a wall must be omitted, and εw is
equal to +1 or −1.
Proof. This corollary is similar to Proposition 4 in [1]. A proof is also similar. We only have
to take into account that signed orbits can be contained in O(λ) ⊗ O±(µ) with the sign “−”.
The corollary is proved. �
Note that in the case of product O(λ) ⊗ O(µ), λ ∈ P+, µ ∈ P+, of usual orbits the orbits
O(ν) with multiplicities mν > 1 may appear in the decomposition . As we see from Corollary,
all coefficients in the decomposition (7.6) are modulo 1. The only problem which appears here
is to find signs of the coefficients εν .
According to Corollary we have
φλ(x)ϕµ(x) =
∑′
w∈W/Wλ
εwϕ|wλ+µ|(x),
where summation is such as in (7.6) and εw are equal to +1 or −1.
Proposition 13. Let O(λ), λ 6= 0, be an orbit and let O±(µ) be a signed orbit. If all elements
wλ+ µ, w ∈W/Wλ, are dominant, then
O(λ)⊗O±(µ) =
⋃′
w∈W/Wλ
O±(wλ+ µ),
where the prime means that terms corresponding to wλ+ µ lying on a wall must be omitted.
Proof. The statement of this proposition follows from the above corollary. �
At the end of this subsection we formulate the following method for decomposition of products
O(λ) ⊗ O±(µ), which follows from statement of Proposition 12. On the first step we shift all
points of the orbit O(λ) by µ. As a result, we obtain the set of points wλ+ µ, w ∈W/Wλ. On
the second step, we map non-dominant elements of this set by elements of the Weyl group W
to the dominant Weyl chamber. On this step we obtain the set |wλ + µ|, w ∈ W/Wλ. Then
according to Proposition 12, O(λ)⊗O±(µ) consists of the signed orbits O±(|wλ+µ|) for which
|wλ + µ| do not lie on a wall of the dominant Weyl chamber. On the third step, we determine
signs of these orbits, taking into account the above propositions or making a direct calculation.
7.3 Products of antisymmetric orbits
Let O±(λ) = {(wλ)detw|w ∈ W} and O±(µ) = {(wµ)detw|w ∈ W} be two signed orbits, where
λ and µ are strictly dominant elements of En. Then
O±(λ)⊗O±(µ) = {(wλ+ w′µ)detww′ |w ∈W,w′ ∈W}
= {(wλ+w1µ)detww1 |w∈W} ∪ {(wλ+w2µ)detww2 |w∈W} ∪ · · ·
∪ {(wλ+wsµ)detwws |w∈W}, (7.7)
42 A. Klimyk and J. Patera
where w1, w2, . . . , ws is the set of all elements of W . Since a product of two signed orbits
decomposes into usual orbits (all points of some of these orbits are taken with the sign “−”),
for decomposition of the product O±(λ) ⊗ O±(µ) into separate orbits it is necessary to take
dominant elements from each term of the right hand side of (7.7). That is, O±(λ)⊗O±(µ) is a
union of the orbits (some of them with the sign “−”) corresponding to points from
D({(wλ+ w1µ)detww1 |w ∈W}), D({(wλ+ w2µ)detww2 |w ∈W}), . . . ,
D({(wλ+ wsµ)detwws |w ∈W}), (7.8)
where D({(wλ + wiµ)detwwi |w ∈ W}) means the set of dominant signed elements in {(wλ +
wiµ)detwwi |w ∈W}.
Proposition 14. The product O±(λ)⊗O±(µ) consists only of orbits (with signs “+” or “−”)
of the form O(|wλ + µ|), w ∈ W , where |wλ + µ| is a dominant weight of the orbit containing
wλ+µ. Moreover, each such orbit O(|wλ+µ|), w ∈W , for which |wλ+µ| does not lie on some
wall of the dominant Weyl chamber, belongs to the product O±(λ)⊗O±(µ).
Proof. For each dominant element wλ+ wiµ from (7.8) there exists an element w′′ ∈ W such
that w′′(wλ + wiµ) = w′λ + µ. It means that wλ + wiµ is of the form |w′λ + µ|, w′ ∈ W , and
the first part of the proposition follows.
Take any element wλ+ µ, w ∈ W , which does not lie on a wall. Then |wλ+ µ| does not lie
on a wall. Then wλ+ µ belongs (with some sign) to the product O±(λ)⊗ O±(µ). This means
that |wλ + µ| also belongs to this product. Therefore, the orbit O(|wλ + µ|) is contained in
O±(λ) ⊗ O±(µ) with some coefficient. This coefficient cannot vanish since if it vanishes, then
in the product O±(λ) ⊗ O±(µ) there are contained points of O±(|wλ + µ|), but taken with an
opposite signs. In this case there exists another element w′λ+µ such that wλ+µ = w′′(w′λ+µ).
Since µ does not lie on a wall, this is not possible. Proposition is proved. �
It follows from Proposition 14 that for decomposition of the product O±(λ) ⊗ O±(µ) into
orbits we have to take all elements wλ + µ, w ∈ W , and to find the corresponding dominant
elements |wλ+ µ|, w ∈W .
Corollary. For the product O±(λ)⊗O±(µ) we have
O±(λ)⊗O±(µ) =
⋃
w∈W
εwO(|wλ+ µ|), (7.9)
where εw is equal to +1 or −1 if |wλ+ µ| does not lie on some wall.
In the case of product O±(λ)⊗O±(µ), λ ∈ P+
+ , µ ∈ P+
+ , of sighed orbits in the decomposition
may appear orbits O(ν) with integral coefficients mν such that mν > 1. Such coefficients may
appear only for ν lying on a wall.
Proposition 15. Let O±(λ) and O±(µ) be two signed orbits. If all elements wλ + µ, w ∈ W ,
are dominant, then
O±(λ)⊗O±(µ) =
⋃
w∈W
εwO(wλ+ µ),
where εw is equal to +1 or −1, if |wλ+ µ| does not lie on some wall.
This proposition is proved in the same way as Proposition 14 and we omit it.
Antisymmetric Orbit Functions 43
Proposition 16. Let O±(λ) and O±(µ) be sighed orbits, and let all elements wλ+ µ, w ∈ W ,
be strictly dominant (that is, they are dominant and do not belong to any wall of the dominant
Weyl chamber). Then
O±(λ)⊗O±(µ) =
⋃
w∈W
εwO(wλ+ µ),
where εw are equal to +1 or −1.
This proposition is proved in the same way as Proposition 2 in [1].
7.4 Decomposition of products for rank 2
We give here examples of decompositions of products of (signed) orbits for the cases A2 and C2.
Orbits for these cases are placed on the plane. Therefore, decompositions can be done by
geometrical calculations on this plane. These cases can be easily considered by using for orbit
points the orthogonal coordinates from Section 3. The corresponding Weyl groups have a simple
description in these coordinates and this gives a possibility to make calculations in a simple
manner.
For the case of A2 at a 6= b we have
A2 : O±(a b)⊗O(c 0) = O±(a+c b) ∪O±(a−c b+c)
∪ −O±(a+b−c c−b) (a > c > b),
O±(a b)⊗O(c 0) = O±(a+c b) ∪O±(a−c b+c)
∪O±(a b−c) (a > c, b > c),
O±(a b)⊗O(c 0) = O±(a+c b) (a = b = c),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(a+b−c c−b)) (a = c > b),
O±(a b)⊗O(c 0) = O±(a+c b) ∪O±(a b−c) (b > a = c),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(c−a a+b) (a < b = c),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(c−a a+b)
∪O±(c−a−b a) (c > a+ b),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(c−a a+b)
∪ −O±(a+b−c c−a) (a+ b > c > b),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(c−a a+b)
∪O±(a b−c) (a+ b > b > c).
If a = b, then we get
O±(a a)⊗O(c 0) = O±(a+c a) ∪O±(c−2a a)
∪ −O±(c−a 2a) (c > 2a),
O±(a a)⊗O(c 0) = O±(a+c a) ∪ −O±(2a−c c−a)
∪ −O±(c−a 2a) (2a > c > a),
O±(a a)⊗O(c 0) = O±(a+c a) ∪O±(a−c a+c)
∪O±(a a−c) (a > c).
Similar products of C2 orbits are of the form
C2 : O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(−a−2b+c b) ∪ −O±(c−a a+b)
∪O±(c−2b−a a+b) (a+b−c<b),
44 A. Klimyk and J. Patera
O±(a b)⊗O(c 0) = O±(a+c b) ∪O±(a+2b−c c−a−b) ∪O±(a−c b+c)
∪ −O±(a+2b−c c−b) (b>c−a−b, a>c),
O±(a b)⊗O(c 0) = O±(a+c b) ∪O±(a+2b−c c−a−b) ∪ −O±(c−a a+b)
∪ −O±(a+2b−c c−b) (b>c−a−b, c>a),
O±(a b)⊗O(c 0) = O±(a+c b) ∪O±(a−c b) ∪O±(a−c b+c)
∪ −O±(a+2b−c c−b) (a+b>b+c),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(c−a a+b−c) ∪ −O±(c−a a+b)
∪ −O±(a+2b−c c−b) (b+c>a+b>c−b),
O±(a b)⊗O(c 0) = O±(a+c b) ∪ −O±(c−a a+b−c) ∪ −O±(c−a a+b)
∪O±(c−a−2b a+b) (a+b<c−b).
The corresponding decompositions of products of antisymmetric and symmetric orbit func-
tions can also be easily written down.
8 Decomposition of antisymmetric W -orbit functions
into antisymmetric W ′-orbit functions
As in Section 7, for these decompositions it is enough to obtain the corresponding decompositions
for signed orbits. For this reason, we shall deal mainly with signed orbits. Our reasoning here
is very similar to that of Section 4 in [1].
8.1 Introduction
Let R be a root system with a Weyl group W , and let R′ be another root system which is
a subset of the set R. Then a Weyl group W ′ for R′ can be considered as a subgroup of W .
Let O±
W (λ) be a signed W -orbit. The set of points of the usual orbit OW (λ) is invariant
with respect to W ′. This means that the signed orbit O±
W (λ) consists of signed W ′-orbits. In
this section we deal with representing O±
W (λ) as a union of signed W ′-orbits. Properties of
such a representation depend on root systems R and R′ (or on Weyl groups W and W ′). We
distinguish two cases:
Case 1: Root systems R and R′ span vector spaces of the same dimension. In this case Weyl
chambers for W are smaller than Weyl chambers for W ′. Moreover, each Weyl chamber for W ′
consists of |W/W ′| chambers for W . Let D+ be a dominant Weyl chamber for the root system R.
Then a dominant Weyl chamber for W ′ consists of W -chambers wiD+, i = 1, 2, . . . , k, k =
|W/W ′|, where wi, i = 1, 2, . . . , k, are representatives of cosets in W/W ′. If λ does not lie on
any wall of the dominant Weyl chamber D+, then
O±
W (λ) =
k⋃
i=1
(detwi) O±
W ′(wiλ), (8.1)
where O±
W ′ are signed W ′-orbits. (Note that if det wi = −1, then (detwi)O±
W ′(wiλ) means the
signed orbit O±
W ′(wiλ) in which each point is taken with opposite sign.)
Representing λ by coordinates in ω-basis it is necessary to take into account that coordinates
of the same point in ω-bases related to the root systems R and R′ are different. There exist
matrices connecting coordinates in these different ω-bases (see [32]).
Antisymmetric Orbit Functions 45
For expanding an antisymmetric W -orbit function into antisymmetric W ′-orbit functions it
is necessary to take into account the formula (8.1). Namely, the following expansion
ϕWλ (x) =
k∑
i=1
(detwi)ϕW
′
wiλ
(x)
corresponds to the decomposition (8.1).
Case 2: Root systems R and R′ span vector spaces of different dimensions. This case is more
complicated. In order to represent O±
W (λ) as a union of signed W ′-orbits, it is necessary to
project points µ of O±
W (λ) to the vector subspace En′ spanned by R′ and to select in the set
of these projected points dominant points with respect to the root system R′. Note that under
projection, different points of O±
W (λ) can give the same point in En′ . This leads to appearing
of coinciding signed W ′-orbits in a representation of O±
W (λ) as a union of W ′-orbits. Moreover,
for some signed W ′-orbits their points must be taken with opposite signs.
As in the previous case, under expansion of an antisymmetric W -orbit function ϕλ(x) into
antisymmetric W ′-orbit functions we have to consider ϕλ(x) on the subspace En′ ⊂ En and to
take into account the corresponding decomposition of the signed orbit O±
W (λ). For this reason,
below in this section we consider decomposition of signedW -orbits intoW ′-orbits. They uniquely
determine the corresponding expansions for antisymmetric orbit functions.
8.2 Decomposition of signed WAn-orbits into WAn−1
-orbits
For such decomposition it is convenient to represent orbit elements in orthogonal coordinates
(see Section 3). Let O±(λ) ≡ O±(m1,m2, . . . ,mn+1) be a signed WAn-orbit with dominant
element λ = (m1,m2, . . . ,mn+1), where
m1 > m2 > · · · > mn > mn+1.
The orthogonal coordinates m1,m2, . . . ,mn+1 satisfy the conditions m1 +m2 + · · ·+mn+1 = 0.
However, we may add to all coordinates mi the same real number, since under this procedure
the ω-coordinates λi = mi −mi+1, i = 1, 2, . . . , n do not change (see Section 3).
The signed orbit O±(λ) consists of all points
w(m1,m2, . . . ,mn+1) = (mi1 ,mi2 , . . . ,min+1), w ∈WAn , (8.2)
where (i1, i2, . . . , in+1) is a permutation of the numbers 1, 2, . . . , n + 1, determined by w. The
sign of (detw) is attached to such point. Points of O±(λ) belong to the vector space En+1. We
restrict these points to the vector subspace En, spanned by the simple roots α1, α2, . . . , αn−1
of An, which form a set of simple roots of An−1. This restriction is reduced to removing the
last coordinate min+1 in points (mi1 ,mi2 , . . . ,min+1) of the signed orbit O±(λ) (see (8.2)). As
a result, we obtain a set of points
(mi1 ,mi2 , . . . ,min) (8.3)
received from the points (8.2). The point (8.3) is dominant if and only if
mi1 ≥ mi2 ≥ · · · ≥ min .
It is easy to see that after restriction to En (that is, under removing the last coordinate) we
obtain from the set of points (8.2) the following set of dominant elements:
(m1, . . . ,mi−1, m̂i,mi+1, . . . ,mn+1), i = 1, 2, . . . , n+ 1,
where a hat over mi means that the coordinate mi must be omitted.
46 A. Klimyk and J. Patera
Thus, the signed WAn-orbit O±(m1,m2, . . . ,mn+1) consists of the following signed WAn−1-
orbits:
O±(m1, . . . ,mi−1, m̂i,mi+1, . . . ,mn+1), i = 1, 2, . . . , n+ 1.
Each of these signed orbits must be taken with a coefficient +1 or −1. Moreover, a coefficient
at the orbit O±(m1, . . . ,mi−1, m̂i,mi+1, . . . ,mn+1) is 1, if after m̂i in the point
(m1, . . . ,mi−1, m̂i,mi+1, . . . ,mn+1)
a number of coordinates is even, and −1 otherwise. This statement completely determines an
expansion of the antisymmetric orbit function ϕ
(WAn )
(m1,m2,...,mn+1) into antisymmetric WAn−1-orbit
function:
ϕ(m1,m2,...,mn+1)(x) =
n+1∑
i=1
(detw(mi))ϕ(m1,...,mi−1,m̂i,mi+1,...,mn+1)(x),
where w(mi) is the permutation which sends the coordinate mi to the end, not changing an
order of other coordinates.
8.3 Decomposition of signed WAn−1
-orbits
into WAp−1
× WAq−1
-orbits, p + q = n
Again we use orthogonal coordinates for orbit elements. We take in the system of simple roots
α1, α2, . . . , αn−1 of An−1 two parts as α1, α2, . . . , αp−1 and αp+1, αp+2, . . . , αp+q−1 ≡ αn−1. The
first part determines WAp−1 and the second part generates WAq−1 . We consider a signed WAn−1-
orbit O±(λ), where λ = (m1,m2, . . . ,mn), m1 > m2 > · · · > mn. This orbit consists of points
w(m1,m2, . . . ,mn) = (mi1 ,mi2 , . . . ,min), w ∈WAn−1 , (8.4)
where (i1, i2, . . . , in) is a permutation of the numbers 1, 2, . . . , n. We restrict points (8.4) to the
vector subspace Ep×Eq spanned by the simple roots α1, α2, . . . , αp−1 and αp+1, αp+2, . . . , αn−1,
respectively. Under restriction the point (8.4) turns into the point
(mi1 ,mi2 , . . . ,mip)(mip+1 ,mip+1 , . . . ,min).
In order to determine a set of signedWAp−1×WAq−1-orbits contained in the signed orbit O±(λ)
we have to choose from (8.4) all elements for which
mi1 > mi2 > · · · > mip , mip+1 > mip+2 > · · · > min .
To find this set of points we have to take all subsets mi1 ,mi2 , . . . ,mip in the set m1,m2, . . . ,mn
such that mi1 > mi2 > · · · > mip . Let Σ denote the collection of such subsets. Then O±(λ)
consists of signed WAp−1 ×WAq−1-orbits
O±((mi1 ,mi2 , . . . ,mip)(mj1 ,mj2 , . . . ,mjq)), (mi1 ,mi2 , . . . ,mip) ∈ Σ, (8.5)
where (mj1 ,mj2 , . . . ,mjq) is a supplement of the subset (mi1 ,mi2 , . . . ,mip) in the whole set
(m1,m2, . . . ,mn), taken in such an order that mj1 > mj2 > · · · > mjq . Each of these WAp−1 ×
WAq−1-orbits is contained in O±(λ) only once. Each such a signed orbit is contained in the
signed orbit O±(λ) with sign “+” if the set of numbers
(mi1 ,mi2 , . . . ,mip ,mj1 ,mj2 , . . . ,mjq)
from (8.5) is obtained from the set (1, 2, . . . , n) by an even permutation and with sign “−”
otherwise. The corresponding expansions of antisymmetric orbit functions ϕ
(WAn−1
)
λ (x) into
antisymmetric WAp−1 ×WAq−1-orbit functions is evident.
Antisymmetric Orbit Functions 47
8.4 Decomposition of signed WBn-orbits into WBn−1
-orbits
and of signed WCn-orbits into WCn−1
-orbits
Decomposition of signed WBn-orbits and decomposition of signed WCn-orbits are fulfilled in the
same way. For this reason, we give a proof only for the case of signed WCn-orbits.
A set of simple roots of Cn consists of roots α1, α2, . . . , αn. The roots α2, . . . , αn constitute
a set of simple roots of Cn−1. They span the subspace En−1.
For determining elements λ of En we use orthogonal coordinates m1,m2, . . . ,mn. Then λ is
strictly dominant if and only if
m1 > m2 > · · · > mn > 0.
Then the signed orbit O±(λ) consists of all points
w(m1,m2, . . . ,mn) = (±mi1 ,±mi2 , . . . ,±min), w ∈WCn , (8.6)
where (i1, i2, . . . , in) is a permutation of the set 1, 2, . . . , n, and all combinations of signs are
possible.
Restriction of elements (8.6) to the vector subspace En−1, defined above, reduces to removing
the first coordinate±mi1 in (8.6). As a result, we obtain from the set of points (8.6) the collection
(±mi2 ,±mi3 , . . . ,±min), w ∈WCn .
Only the points (mi2 ,mi3 , . . . ,min−1 ,min) with positive coordinates may be dominant. More-
over, such a point is dominant if and only if
mi2 > mi3 > · · · > min .
Therefore, under restriction of points (8.6) to En−1 we obtain the following strictly WCn−1-
dominant elements:
(m1,m2, . . . ,mi−1, m̂i,mi+1, . . . ,mn), i = 1, 2, . . . , n, (8.7)
where a hat over mi means that the coordinate mi must be omitted. Moreover, the element (8.7)
with fixed i can be obtained from two elements in (8.6), namely, from (m1,m2, . . . ,mi−1,±mi,
mi+1, . . . ,mn). In the signed orbit O±(m1,m2, . . . ,mn) these two elements have opposite signs.
Thus, the signed WCn-orbits O±(m1,m2, . . . ,mn) consists of the following signed WCn−1-
orbits:
O±(m1,m2, . . . ,mi−1, m̂i,mi+1, . . . ,mn), i = 1, 2, . . . , n.
Each such signed WCn−1-orbit is contained in O±(m1,m2, . . . ,mn) twice with opposite signs.
Therefore, a restriction of the antisymmetric orbit function ϕ(m1,m2,...,mn) to the subspa-
ce En−1, described above, vanishes.
For WBn-orbits we have similar assertions. A signed WBn-orbits O±(m1,m2, . . . ,mn), m1 >
m2 > · · · > mn > 0, consists of WBn−1-orbits
O±(m1,m2, . . . ,mi−1, m̂i,mi+1, . . . ,mn), i = 1, 2, . . . , n,
and each such orbit is contained in the decomposition two times (with opposite signs), that
is, a restriction of the antisymmetric orbit function ϕ(m1,m2,...,mn) of Bn to the subspace En−1,
described above, vanishes.
48 A. Klimyk and J. Patera
8.5 Decomposition of signed WCn-orbits into WAp−1
× WCq-orbits, p + q = n
If α1, α2, . . . , αn are simple roots for Cn, then α1, α2, . . . , αp−1 are simple roots for Ap−1 (they
can be embedded into the linear subspace Ep) and αp+1, αp+2, . . . , αn are simple roots for Cq
(they generate the linear subspace Eq).
We use orthogonal coordinates for elements of En and consider a signed WCn-orbit O±(λ),
m1 > m2 > · · · > mn > 0. This orbit consists of all points (8.6). Restriction of these points to
the vector subspace Ep × Eq reduces to splitting of coordinates (8.6) into two parts:
(±mi1 ,±mi2 , . . . ,±mip)(±mip+1 , . . . ,±min). (8.8)
Due to the condition m1 > m2 > · · · > mn > 0, these elements do not lie on walls of the
WAp−1 ×WCq -chambers. We have to choose dominant elements (with respect to the Weyl group
WAp−1 ×WCq) in the set of points (8.8). The conditions of dominantness for elements of Ep
and Eq show that only the elements
(mi1 , . . . ,mij ,−mij+1 · · · ,−mip)(mip+1 , . . . ,min), j = 0, 1, 2, . . . , p,
satisfying the conditions
mi1 > mi2 > · · · > mij , mij+1 < mij+2 < · · · < mip , mip+1 > mip+2 > · · · > min ,
are dominant. Moreover, each such point is contained in the signed WCn-orbit O±(λ) only once.
This assertion completely determines a list of signed WAp−1×WCq -orbits in O±(λ). Each signed
WAp−1 ×WCq -orbit is contained in O±(λ) only once.
This assertion uniquely determines a list of antisymmetric WAp−1 × WCq -orbit functions,
contained in the antisymmetric WCn-orbit function ϕλ. However, it is necessary to determine
signs of antisymmetric WAp−1 ×WCq -orbit functions in the decomposition. It is easily made by
using the description of the Weyl groups in the Euclidean space En with orthogonal coordinates.
8.6 Decomposition of signed WDn-orbits into signed WDn−1
-orbits
Assume that α1, α2, . . . , αn is the set of simple roots of Dn, n > 4. Then α2, . . . , αn are simple
roots of Dn−1. The last roots span the subspace En−1.
For elements λ of En we again use orthogonal coordinates m1,m2, . . . ,mn. Then λ is strictly
dominant if and only if m1 > m2 > · · · > mn−1 > |mn|. We assume that λ satisfies the condition
m1 > m2 > · · · > mn > 0.
Then the signed orbit O±(λ) consists of all points
w(m1,m2, . . . ,mn) = (±mi1 ,±mi2 , . . . ,±min), w ∈WDn , (8.9)
where (i1, i2, . . . , in) is a permutation of the numbers 1, 2, . . . , n and there exists an even number
of signs “−”. Restriction of elements (8.9) to the subspace En−1 reduces to removing the first
coordinate ±mi1 in (8.9). As a result, we obtain from the set of points (8.9) the collection
(±mi2 ,±mi3 , . . . ,±min), w ∈WDn ,
where a number of signs “−” may be either even or odd. Only points of the form (mi2 ,mi3 , . . . ,
min−1 ,±min) may be dominant. Moreover, such a point is dominant if and only if
mi2 > mi3 > · · · > min−1 > |min |.
Antisymmetric Orbit Functions 49
Therefore, under restriction of points (8.9) to En−1 we obtain the following WDn−1-dominant
elements:
(m1,m2, . . . ,mi−1, m̂i,mi+1, . . . ,mn−1,±mn), i = 1, 2, . . . , n, (8.10)
where a hat over mi means that the coordinate mi must be omitted. Moreover, the ele-
ment (8.10) with fixed i can be obtained only from one element in (8.9), namely, from element
(m1,m2, . . . ,mi−1,±mi,mi+1, . . . ,±mn), where at mi and mn signs are coinciding.
Thus, the signed WDn-orbit O±(m1,m2, . . . ,mn) with m1 > m2 > · · · > mn > 0 consists of
the following signed WDn−1-orbits:
O±(m1,m2, . . . ,mi−1, m̂i,mi+1, . . . ,±mn), i = 1, 2, . . . , n.
Each such signed WDn−1-orbit is contained in O±(m1,m2, . . . ,mn) only once (with sign “+” or
“−”). A sign of such an orbit depends on a number i and on a sign at mn. This sign is uniquely
determined by the sign (detw) of the corresponding element w of WDn .
It is shown similarly that the signed WDn-orbit
O±(m1, . . . ,mn−1,−mn), m1 > m2 > · · · > mn > 0
consists of the same signed WDn−1-orbits as the WDn-orbits O±(m1, . . . ,mn−1,mn) with the
same numbers m1, . . . ,mn−1,mn does.
The above assertions uniquely determine expansions for the corresponding antisymmetric
WDn-orbit functions.
8.7 Decomposition of signed WDn-orbits into WAp−1
× WDq-orbits,
p + q = n, q ≥ 4
If α1, α2, . . . , αn are simple roots for Dn, then α1, α2, . . . , αp−1 are simple roots for Ap−1 (they
can be embedded into the Euclidean subspace Ep) and αp+1, αp+2, . . . , αn are simple roots for Dq
(they generate the Euclidean subspace Eq).
We use orthogonal coordinates in En and consider signed WDn-orbits O±(λ) with λ =
(m1,m2, . . . ,mn) such thatm1 > m2 > · · · > mn > 0. The orbit O(λ) consists of all points (8.9).
Restriction of these points to the vector subspace Ep−1×Eq reduces to splitting the set of coor-
dinates (8.9) into two parts:
(±mi1 ,±mi2 , . . . ,±mip)(±mip+1 , . . . ,±min). (8.11)
Due to the condition m1 > m2 > · · · > mn > 0, these elements do not lie on walls of the
WAp−1 ×WDq -chambers. We have to choose dominant elements (with respect to the Weyl group
WAp−1 ×WDq) in the set of points (8.11). The conditions of dominantness for elements of Ep
and Eq show that only the elements
(mi1 , . . . ,mij ,−mij+1 . . . ,−mip)(mip+1 , . . . ,±min), j = 0, 1, 2, . . . , p,
having even number of sign minus and satisfying the conditions
mi1 > mi2 > · · · > mij , mij+1 < mij+2 < · · · < mip , mip+1 > mip+2 > · · · > min ,
are dominant. Moreover, each such point is contained in the WDn-orbit O±(λ) only once. These
assertions completely determine a list of signed WAp−1 ×WDq -orbits in the signed WDn-orbit
O±(λ). All signed WAp−1 ×WDq -orbits are contained in O±(λ) with multiplicity 1 and with
sign “+” or “−”. This determines uniquely expansions for the corresponding antisymmetric
WDn-orbit functions.
50 A. Klimyk and J. Patera
9 Characters of representations and antisymmetric
orbit functions
Antisymmetric orbit functions ϕλ(x) with λ ∈ P+
+ are closely related to characters of irreducible
representations of the corresponding compact Lie group G. This relation serves for derivations
of some properties of antisymmetric orbit functions.
9.1 Connection of characters with orbit functions
To each Coxeter–Dynkin diagram there corresponds a connected compact semisimple Lie
group G. Let us fix a Coxeter–Dynkin diagram and, therefore, a connected compact Lie group G.
A complex valued function f(g) on G satisfying the condition
f(g) = f(hgh−1), h ∈ G,
is called a class function. It is constant on classes of conjugate elements.
For simplicity, we assume that G is realized by matrices such that the set of its diagonal
matrices constitute a Cartan subgroup, which will be denoted by H. This subgroup can be
identified with the n-dimensional torus T, where n is a rank of the group G. The subgroup H
can be represented as H = exp(ih), where h is a real form of an appropriate Cartan subalgebra
of the complex semisimple Lie algebra, determined by the Coxeter–Dynkin diagram.
It is well-known that each element g of G is conjugate to some element of H, that is, class
functions are uniquely determined by their values on H.
There exists a one-to-one correspondence between irreducible unitary representations of the
group G and integral highest weights λ ∈ P+, where P+ is determined by the Coxeter–Dynkin
diagram. The irreducible representation, corresponding to a highest weight λ, will be denoted
by Tλ. The representation Tλ and its properties are determined by its character χλ(g), which is
defined as the trace of Tλ(g):
χλ(g) = Tr Tλ(g), g ∈ G.
Since Tr Tλ(hgh−1) = Tr Tλ(g), h ∈ G, the character χλ is a class function, that is, it is uniquely
determined by its values on the subgroup H.
All the operators Tλ(h), h ∈ H, are diagonal with respect to an appropriate basis of the
representation space (this basis is called a weight basis) and their diagonal matrix elements are
of the form e2πi〈µ,x〉, where µ ∈ P is a weight of the representation Tλ, x = (x1, x2, . . . , xn) are
coordinates of an element t of the Cartan subalgebra h in an appropriate coordinate system
(that is, coordinates on the torus T) and 〈·, ·〉 is an appropriate bilinear form, which can be
chosen coinciding with the scalar product on En, considered above. Then the character χλ(h)
is a linear combination of the diagonal matrix elements:
χλ(h) =
∑
µ∈D(λ)
cµλe
2πi〈µ,x〉, (9.1)
where D(λ) is the set of all weights of the irreducible representation Tλ and cµλ is a multiplicity
of the weight µ ∈ D(λ) in the representation Tλ. It is known from representation theory that
the weight system D(λ) of Tλ is invariant with respect to the Weyl group W , corresponding to
the Coxeter–Dynkin diagram, and cwµλ = cµλ, w ∈ W , for each µ ∈ D(λ). This means that the
character χλ(h) can be represented as
χλ(h) =
∑
µ∈D+(λ)
cµλφµ(x), (9.2)
Antisymmetric Orbit Functions 51
where D+(λ) is the set of all dominant weights in D(λ) and φµ(x) is a symmetric orbit function,
corresponding to the weight µ ∈ D+(λ). Representing χλ(h) as χλ(x), where x = (x1, x2, . . . , xn)
are coordinates corresponding to the element t ∈ h such that h = exp 2πit, we can make an
analytic continuation of both sides of (9.2) to the n-dimensional Euclidean space En. Since the
right hand side of (9.2) is invariant under transformations from the affine Weyl group W aff ,
corresponding to the Weyl group W , the function χλ(x) is also invariant under the affine Weyl
group W aff . That is, it is enough to define χλ(x) only on the fundamental domain F of the
group W aff . To this fundamental domain F there corresponds a fundamental domain (we denote
it by F̃ ) in the subgroup H (and on the torus T).
Many properties of orbit functions follow from properties of characters χλ, and we consider
them as known from representation theory.
The well-known Weyl formula for characters of irreducible representations of the group G
states [33] that under appropriate selection of coordinates (x1, x1, . . . , xn) of points h ∈ H we
have
χλ(h) =
∑
w∈W
(detw)e2πi〈λ+ρ,x〉∑
w∈W
(detw)e2πi〈ρ,x〉 =
ϕλ+ρ(x)
ϕρ(x)
, (9.3)
where, as before, ρ = 1
2
∑
α>0
α. This is why antisymmetric orbit functions are so important for
representation theory. Note that for dominant λ ∈ P+ the element λ + ρ is strictly dominant;
for this reason, the antisymmetric orbit function ϕλ+ρ(x) in (9.3) does not vanish for λ ∈ P+.
9.2 Orthogonality of characters
The relation (9.3) between characters of irreducible representations of compact Lie groups and
the corresponding antisymmetric orbit functions leads to a simple proof of orthogonality of
irreducible characters χλ(h) ≡ χλ(x). Indeed, due to the orthogonality (5.2) for antisymmetric
orbit functions and to the relation (9.3) we have∫
F
ϕλ+ρ(x)ϕλ′+ρ(x)dx =
∫
F
χλ(x)χλ′(x)|ϕρ(x)|2dx = δλλ′ ,
that is, irreducible characters are orthogonal on F with respect to the measure |ϕρ(x)|2dx. The
expressions for the function ϕρ(x) for the classical compact Lie groups are given in Subsection 5.9.
9.3 Relations for antisymmetric orbit functions
The formulas (9.2) and (9.3) are a source of relations for antisymmetric orbit functions.
Comparing the expressions (9.2) and (9.3) for characters, one gets the relation
ϕλ+ρ(x) =
∑
µ∈D+(λ)
cµλφµ(x)ϕρ(x), (9.4)
where, as before, cµλ are multiplicities of weights µ in the irreducible representation Tλ of G.
Let Tλ and Tµ be two irreducible representations of G. Then their tensor product decomposes
as a direct sum of irreducible representations of G as
Tλ ⊗ Tµ =
∑
ν∈P+
mλµ
ν Tν , (9.5)
52 A. Klimyk and J. Patera
where mλµ
ν is a multiplicity of the irreducible representation Tν in the tensor product. Since Tλ
and Tµ are finite dimensional representations, the sum on the right hand side of (9.5) is finite.
To the decomposition (9.5) there corresponds a relation for characters,
χλ(x)χµ(x) =
∑
ν∈P+
mλµ
ν χν(x). (9.6)
Due to (9.3) it can be written as
ϕλ+ρ(x)
ϕρ(x)
ϕµ+ρ(x)
ϕρ(x)
=
∑
ν∈P+
mλ
µν
ϕν+ρ(x)
ϕρ(x)
,
where summation is such as in (9.6). Therefore, we have the following expression for a product
of two antisymmetric orbit functions:
ϕλ+ρ(x)ϕµ+ρ(x) =
∑
ν∈P+
mλµ
ν ϕν+ρ(x)ϕρ(x). (9.7)
In particular, if µ = ωi, where ωi is i-th fundamental wight of the group G, then this formula
takes the form
ϕλ+ρ(x)ϕωi+ρ(x) =
∑
ν∈P+
mλωi
ν ϕν+ρ(x)ϕρ(x).
Since multiplicities of irreducible constituents in the tensor product Tλ ⊗ Tωi for many groups
can be found in a simple way (in many cases these multiplicities are equal to 1; for example, if
i = 1, then for the groups SU(n) and SO(n) they do not exceed 1), then this formula can be
considered as a recurrence relation for antisymmetric orbit functions.
10 Antisymmetric orbit function transforms
As in the case of symmetric orbit functions, antisymmetric orbit functions determine certain
orbit function transforms which generalize the sine transform (in the case of symmetric orbit
functions these transforms generalize the cosine transform) [29, 31].
As in the case of symmetric orbit functions, antisymmetric orbit functions determine three
types of orbit function transforms: the first one is related to the antisymmetric orbit func-
tions ϕλ(x) with integral λ, the second one is related to ϕλ(x) with dominant λ ∈ En, and the
third one is the related discrete transforms.
10.1 Decomposition in antisymmetric orbit functions on F
Let f(g) be a continuous class function on G (see Subsection 9.1). It defines a continuous
function on the commutative subgroup H. We assume that this function on H has continuous
partial derivatives of all orders with respect to analytic parameters on H. Such function f can
be decomposed in characters of irreducible unitary representations of G:
f(h) =
∑
λ∈P+
cλχλ(h). (10.1)
We see from this decomposition that each class function is symmetric with respect to the corre-
sponding affine Weyl group W aff (since characters χλ admit this symmetry) and, therefore, is
uniquely determined by its values on the fundamental domain F̃ in H.
Antisymmetric Orbit Functions 53
Going to the coordinate description of the points h ∈ H (see Subsection 9.1), we obtain
f(x) =
∑
λ∈P+
cλχλ(x), x ∈ En.
Taking into account formula (9.3) for the characters we receive
ϕρ(x)f(x) =
∑
λ∈P+
cλϕλ+ρ(x). (10.2)
Due to Proposition 7, we may state that any antisymmetric (with respect to the affine Weyl
group W aff) continuous function f on En, which has continuous derivatives and vanishes on the
boundary ∂F of the fundamental domain F , can be represented in the form f(x) = ϕρ(x)f̃(x),
where f̃(x) is a symmetric (with respect to W aff) continuous function on En with continuous
derivatives. Thus, due to (10.2) we may state that any antisymmetric (with respect to W aff)
continuous function f on En, which has continuous derivatives and vanishes on the boundary ∂F ,
can be decomposed in antisymmetric orbit functions ϕλ, λ ∈ P+
+ ,
f(x) =
∑
λ∈P+
+
cλϕλ(x). (10.3)
By the orthogonality relation (5.2) for antisymmetric orbit functions, the coefficients cλ in
this decomposition are determined by the formula
cλ =
∫
F
f(x)ϕλ(x)dx. (10.4)
Moreover, the Plancherel formula∑
λ∈P+
|cλ|2 =
∫
F
|f(x)|2dx (10.5)
holds, which means that the Hilbert spaces with the appropriate scalar products are isometric.
Formula (10.4) is the antisymmetrized Fourier transform of the function f(x). Formula (10.3)
gives an inverse transform. Formulas (10.3) and (10.4) give the orbit function transforms corre-
sponding to antisymmetric orbit functions ϕλ, λ ∈ P+
+ .
Let L2
0(F ) denote the Hilbert space of functions on the fundamental domain F , which vanish
on the boundary ∂F of the fundamental domain, with the scalar product
〈f1, f2〉 =
∫
F
f1(x)f2(x)dx.
The set of continuous functions on F (vanishing on the boundary ∂F ) with continuous derivatives
is dense in L2
0(F ). Therefore, the formulas (10.3)–(10.5) show that the set of orbit functions ϕλ,
λ ∈ P+
+ , form an orthogonal basis of L2
0(F ).
10.2 Symmetric and antisymmetric multivariate sine and cosine series
Symmetric and antisymmetric orbit functions for the Coxeter–Dynkin diagram Cn can be ex-
pressed in terns of symmetric and antisymmetric multivariate sine and cosine functions (see
formulas (4.19) and (5.4)). Their application in the formulas for symmetric and antisymmetric
orbit function transforms gives antisymmetric and symmetric multivariate sine and cosine series
expansions.
54 A. Klimyk and J. Patera
The formulas (10.3) and (10.4), applied to the case Cn, determine expansions of functions,
given on the fundamental domain
F = {1/2 > x1 > x2 > · · · > xn > 0}
for the Coxeter–Dynkin diagram Cn, into antisymmetric multivariate sine functions:
f(x) =
∑
m∈P+
+
cm det (sin 2πmixj)
n
i,j=1 , (10.6)
where m = (m1,m2, . . . ,mn) are integer n-tuples such that m1 > m2 > · · · > mn > 0, and the
coefficients cm are given by the formula
cm = 22n
∫
F
f(x) det (sin 2πmixj)
n
i,j=1 dx. (10.7)
The Plancherel formula is of the form∑
m∈P+
+
|cm|2 = 22n
∫
F
f(x)|f(x)|2dx.
Similarly, using the symmetric orbit function transform on the fundamental domain F (see
Subsection 8.2 in [1]), determined by symmetric orbit functions for the Coxeter–Dynkin diag-
ram Cn given by the formulas (4.24) and (5.4), one obtains symmetric multivariate cosine ex-
pansion on F :
f(x) =
∑
m∈P+
cmdet+(cos 2πmixj)ni,j=1, (10.8)
where m = (m1,m2, . . . ,mn) are integer n-tuples such that m1 ≥ m2 ≥ · · · ≥ mn ≥ 0, and the
coefficients cm are given by the formula
cm = 22n
∫
F
f(x)det+(cos 2πmixj)ni,j=1dx. (10.9)
The Plancherel formula is of the form∑
m∈P+
|cm|2 = 22n
∫
F
f(x)|f(x)|2dx.
10.3 Orbit function transform on the dominant Weyl chamber
The expansion (10.3) of functions on the fundamental domain F is an expansion in the antisym-
metric orbit functions ϕλ(x) with integral strictly dominant weights λ. The antisymmetric orbit
functions ϕλ(x) with λ lying in the dominant Weyl chamber (and not obligatory integral) are
not invariant with respect to the corresponding affine Weyl group W aff . They are invariant only
with respect to the Weyl group W . A fundamental domain of W coincides with the dominant
Weyl chamber D+. For this reason, the orbit functions ϕλ(x), λ ∈ D+
+, determine another orbit
function transform (a transform on D+).
We began with the usual Fourier transforms on Rn:
f̃(λ) =
∫
Rn
f(x)e2πi〈λ,x〉dx, (10.10)
f(x) =
∫
Rn
f̃(λ)e−2πi〈λ,x〉dλ. (10.11)
Antisymmetric Orbit Functions 55
Let the function f(x) be anti-invariant with respect to the Weyl group W , that is, f(wx) =
(detw)f(x), w ∈W . It is easy to check that the function f̃(λ) is also anti-invariant with respect
to the Weyl group W . Replace in (10.10) λ by wλ, w ∈ W , multiply both sides by detw, and
sum these both side over w ∈W . Then instead of (10.10) we obtain
f̃(λ) =
∫
D+
f(x)ϕλ(x)dx, λ ∈ D+
+, (10.12)
where we have taken into account that f(x) is anti-invariant with respect to W .
Similarly, starting from (10.11), we obtain the inverse formula:
f(x) =
∫
D+
f̃(λ)ϕλ(x)dλ. (10.13)
For the transforms (10.12) and (10.13) the Plancherel formula∫
D+
|f(x)|2dx =
∫
D+
|f̃(λ)|2dλ
holds.
10.4 Symmetric and antisymmetric multivariate sine
and cosine integral transforms
The orbit function transforms (10.12) and (10.13) in the case of the Coxeter–Dynkin diagram Cn
lead to symmetric and antisymmetric multivariate sine and cosine integral transforms.
Taking into account the expression (4.19) for antisymmetric orbit functions for Cn we obtain
the transform
f̃(λ) =
∫
D+
f(x) det (sin 2πλixj)
n
i,j=1 dx, (10.14)
where λ = (λ1, λ2, . . . , λn) ∈ D+
+, that is, λ1 > λ2 > · · · > λn > 0 (the function f̃(λ) vanishes
on the boundary of D+). The inverse transform is of the form
f(x) = 22n
∫
D+
f̃(λ) det (sin 2πλixj)
n
i,j=1 dλ. (10.15)
For these transforms the Plancherel formula∫
D+
|f(x)|2dx = 22n
∫
D+
|f̃(λ)|2dλ
holds.
Similar transformations hold for symmetric multivariate cosine function:
f̃(λ) =
∫
D+
f(x)det+ (cos 2πλixj)
n
i,j=1 dx, (10.16)
f(x) = 22n
∫
D+
f̃(λ)det+ (cos 2πλixj)
n
i,j=1 dλ, (10.17)
where λ = (λ1, λ2, . . . , λn) ∈ D+ and the function det+ (cosλixj)
n
i,j=1 is given by formula (5.4).
56 A. Klimyk and J. Patera
11 Multivariate generalization of the finite Fourier transform
and of finite sine and cosine transforms
Along with the integral Fourier transform there exists a discrete Fourier transform. Similarly, it
is possible to introduce a finite orbit function transform, based on antisymmetric orbit functions.
It is done in the same way as in the case of symmetric orbit functions in [1] by using the results
of the paper [21] (see [31]). We first consider the finite Fourier transform. Then we consider a
general theory (appropriate for any connected Coxeter–Dynkin diagram). In the last subsections
we give concrete antisymmetric and symmetric generalizations of the finite Fourier transform.
In particular, we consider antisymmetric and symmetric multivariate finite Fourier transforms,
discrete sine and cosine transforms, and antisymmetric and symmetric multivariate discrete sine
and cosine transforms.
11.1 Finite Fourier transform
Let us fix a positive integer N and consider the numbers
emn := N−1/2 exp(2πimn/N), m, n = 1, 2, . . . , N. (11.1)
The matrix (emn)Nm,n=1 is unitary, that is,∑
k
emkenk = δmn,
∑
k
ekmekn = δmn. (11.2)
Indeed, according to the formula for a sum of a geometric progression we have
ta + ta+1 + · · ·+ ta+r = (1− t)−1ta(1− tr+1), t 6= 1,
ta + ta+1 + · · ·+ ta+r = r + 1, t = 1.
Setting t = exp(2πi(m− n)/N), a = 1 and r = N − 1, we prove (11.2).
Let f(n) be a function of n ∈ {1, 2, . . . , N}. We may consider the transform
N∑
n=1
f(n)emn ≡ N−1/2
N∑
n=1
f(n) exp(2πimn/N) = f̃(m). (11.3)
Then due to unitarity of the matrix (emn)Nm,n=1, we express f(n) as a linear combination of
conjugates of the functions (11.1):
f(n) = N−1/2
N∑
m=1
f̃(m) exp(−2πimn/N). (11.4)
The function f̃(m) is a finite Fourier transform of f(n). This transform is a linear map. The
formula (11.4) gives an inverse transform. The Plancherel formula
N∑
m=1
|f̃(m)|2 =
N∑
n=1
|f(n)|2
holds for transforms (11.3) and (11.4). This means that the finite Fourier transform preserves
the norm introduced in the space of functions on {1, 2, . . . , N}.
The finite Fourier transform on the r-dimensional linear space Er is defined similarly. We
again fix a positive integer N . Let m = (m1,m2, . . . ,mr) be an r-tuple of integers such that
Antisymmetric Orbit Functions 57
each mi runs over the integers 1, 2, . . . , N . Then the finite Fourier transform on Er is given by
the kernel
emn := em1n1em2n2 · · · emrnr = N−r/2 exp(2πim · n/N),
where m · n = m1n1+m2n2+· · ·+mrnr. If F (m) is a function of r-tuples m, mi ∈ {1, 2, . . . , N},
then the finite Fourier transform of F is given by
F̃ (n) = N−r/2
∑
m
F (m) exp(2πim · n/N).
The inverse transform is
F (m) = N−r/2
∑
n
F̃ (n) exp(−2πim · n/N).
The corresponding Plancherel formula is of the form
∑
m
|F (m)|2 =
∑
n
|F̃ (n)|2.
11.2 W -invariant lattices
In order to determine an analogue of the finite Fourier transform, based on antisymmetric orbit
functions, we need an analogue of the set
{m = {m1,m2, . . . ,mn} | mi ∈ {1, 2, . . . , N}},
used for multidimensional finite Fourier transform. Such a set has to be invariant with respect
to the Weyl group W (see [21]).
We know that Q∨ is a discrete W -invariant subset of En. Clearly, the set 1
mQ
∨ is also
W -invariant, where m is a fixed positive integer. Then the set
Tm = 1
mQ
∨/Q∨
is finite and W -invariant. If α1, α2, . . . , αl is the set of simple root for the Weyl group W , then
Tm can be identified with the set of elements
m−1
l∑
i=1
diα
∨
i , di = 0, 1, 2, . . . ,m− 1. (11.5)
We select from Tm the set of elements which belongs to the fundamental domain F . These
elements lie in the collection 1
mQ
∨ ∩ F .
Let µ ∈ 1
mQ
∨∩F be an element determining an element of Tm and let M be the least positive
integer such that Mµ ∈ P∨. Then there exists the least positive integer N such that Nµ ∈ Q∨.
One has M |N and N |m.
The collection of points of Tm which belong to F (we denote the set of these points by FM )
coincides with the set of elements
s =
s1
M
ω∨1 + · · ·+ sl
M
ω∨l , ω∨i =
2ωi
〈αi, αi〉
, (11.6)
where s1, s2, . . . , sl runs over values from {0, 1, 2, . . . } and satisfy the following condition: there
exists a non-negative integer s0 such that
s0 +
l∑
i=1
simi = M, (11.7)
58 A. Klimyk and J. Patera
where m1,m2, . . . ,ml are positive integers from formula (2.8). Values of mi for all simple Lie
algebras can be found in Subsection 2.4.
Indeed, the fundamental domain consists of all points y from the dominant Weyl chamber for
which 〈y, ξ〉 ≤ 1, where ξ is the highest (long) root, ξ =
l∑
i=1
miαi. Since for elements s of (11.6)
one has si/M ≥ 0 and
〈s, ξ〉 =
1
M
l∑
i=1
simi =
1
M
(M − s0) ≤ 1,
then s ∈ F . The converse reasoning shows that points of 1
mQ
∨ ∩ F must be of the form (11.6).
To every positive integer M there corresponds the grid FM of points (11.6) in F which
corresponds to some set Tn such that M |m. The precise relation between M and n can be
defined by the grid FM (see [21]). Acting upon the grid FM by elements of the Weyl group W
we obtain the whole set Tm.
Since antisymmetric orbit functions vanish on the boundary of a fundamental domain F , it
make sense to consider also a subgrid F−
M consisting of all points of FM which do not lie on
a wall of F .
Example. Grids FM for A1. We take into account results of Example in Subsection 4.1. For A1
we have ω∨ = ω = α/2, where α is the simple root. Elements of P+ coincide with mω, m ∈ Z+.
Fixing M ∈ Z+ we have
FM =
{
s = s1
M , where s0 + s1 = M for s0, s1 ∈ Z≥0
}
.
Therefore,
FM =
{
0, 1
M ,
2
M , , . . . ,
M−1
M , 1
}
.
11.3 Grids FM for A2, C2 and G2
In this section we give some examples of grids FM for the rank two cases (see [14] and [15]).
Since the long root ξ of A2 is representable in the form ξ = α1 +α2, where α1 and α2 are simple
roots, that is, m1 = m2 = 1 (see formula (11.7)), then
FM (A2) =
{
s1
M ω1 + s2
M ω2; s0 + s1 + s2 = M, s0, s1, s2 ∈ Z≥0
}
.
It is seen from here that the vertices 0, ω1, ω2 of the fundamental domain F (A2) belong to each
grid FM (A2). A direct computation shows that in the ω-coordinates we have
F2(A2) =
{
(0, 0), (1, 0), (0, 1),
(
1
2 , 0
)
,
(
0, 1
2
)
,
(
1
2 ,
1
2
)}
,
F3(A2) =
{
(0, 0), (1, 0), (0, 1),
(
1
3 , 0
)
,
(
0, 1
3
)
,
(
2
3 , 0
)
,
(
0, 2
3
)
,
(
2
3 ,
1
3
)
,
(
1
3 ,
2
3
)
,
(
1
3 ,
1
3
)}
.
In F2(A2), only the point
(
1
2 ,
1
2
)
does not belong to a wall of the fundamental domain F (A2).
In F3(A2), three points
(
1
3 ,
2
3
)
,
(
2
3 ,
1
3
)
,
(
1
3 ,
1
3
)
do not belong to a wall of F (A2).
The set F−
5 (A2) consists of the points
F−
5 (A2) =
{(
1
5 ,
3
5
)
,
(
2
5 ,
2
5
)
,
(
3
5 ,
1
5
)
,
(
1
5 ,
2
5
)
,
(
2
5 ,
1
5
)
,
(
1
5 ,
1
5
)}
.
Since the long root ξ of C2 is representable in the form ξ = 2α1 + α2, where α1 and α2 are
simple roots, that is, m1 = 2,m2 = 1, then
FM (C2) =
{
s1
M ω
∨
1 + s2
M ω
∨
2 ; s0 + 2s1 + s2 = M, s0, s1, s2 ∈ Z≥0
}
.
Antisymmetric Orbit Functions 59
A direct computation shows that in the ω∨-coordinates we have
F2(C2) =
{
(0, 0), (0, 1),
(
1
2 , 0
)
,
(
0, 1
2
)}
,
F3(C2) =
{
(0, 0), (0, 1),
(
1
3 , 0
)
,
(
0, 1
3
)
,
(
0, 2
3
)
,
(
1
3 ,
1
3
)}
,
F−
7 (C2) =
{(
1
7 ,
4
7
)
,
(
2
7 ,
2
7
)
,
(
1
7 ,
3
7
)
,
(
2
7 ,
1
7
)
,
(
1
7 ,
2
7
)
,
(
1
7 ,
1
7
)}
.
Since the long root ξ of G2 is representable in the form ξ = 2α1 + 3α2, where α1 and α2 are
simple roots, that is, m1 = 2, m2 = 3, then
FM (G2) =
{
s1
M ω
∨
1 + s2
M ω
∨
2 ; s0 + 2s1 + 3s2 = M, s0, s1, s2 ∈ Z≥0
}
.
A computation shows that in the ω∨-coordinates we have
F2(G2) = {(0, 0), (1, 0)} ,
F3(G2) =
{
(0, 0),
(
0, 1
3
)
,
(
1
3 , 0
)}
,
F4(G2) = F2(G2) ∪
{(
1
4 , 0
)
,
(
0, 1
4
)}
,
F5(G2) =
{
(0, 0),
(
0, 1
5
)
,
(
1
5 , 0
)
,
(
1
5 ,
1
5
)
,
(
2
5 , 0
)}
,
F8(G2) = F4(G2) ∪
{(
1
8 , 0
)
,
(
0, 1
8
)
,
(
1
8 ,
1
8
)
,
(
1
4 ,
1
8
)}
,
F−
14(G2) =
{(
1
7 ,
3
14
)
,
(
5
14 ,
1
14
)
,
(
3
14 ,
1
7
)
,
(
1
14 ,
3
14
)
,
(
2
7 ,
1
14
)
,
(
1
7 ,
1
7
)
,(
3
14 ,
1
14
)
,
(
1
14 ,
1
7
)
,
(
1
7 ,
1
14
)
,
(
1
14 ,
1
14
)}
.
11.4 Expanding in antisymmetric orbit functions through finite sets
Let us give an analogue of the finite Fourier transform when instead of exponential functions we
use antisymmetric orbit functions. This analogue is not so simple as finite Fourier transform.
For this reason, we consider some weak form of the transform. In fact, we consider this weak
form in order to be able to recover (at least approximately) the decomposition f(x) =
∑
λ
aλϕλ(x)
by values of f(x) on a finite set of points.
Considering the finite Fourier transform in Section 11.1, we have restricted the exponential
function to a discrete set. Similarly, in order to determine a finite transform, based on anti-
symmetric orbit functions, we have to restrict orbit functions ϕλ(x) to appropriate finite sets of
values of x. Candidates for such finite sets are sets Tm. However, antisymmetric orbit functions
ϕλ(x) with integral λ are invariant with respect to the affine Weyl group W aff . For this reason,
we consider orbit functions ϕλ(x) on grids FM .
On the other side, we also have to choose a finite number of antisymmetric orbit functions,
that is, a finite number of integral strictly dominant elements λ. The best choice is when
a number of orbit functions coincides with |FM |. These antisymmetric orbit functions must be
selected in such a way that the matrix
(ϕλi
(xj))λi∈Ω,xj∈FM
(11.8)
(where Ω is our finite set of strictly dominant elements λ) is not singular. In order to have non-
singularity of this matrix some conditions must be satisfied. In general, they are not known.
For this reason, we consider some, more weak, form of the transform (when |Ω| ≥ |FM |) and
then explain how the set |Ω| of λ ∈ P+
+ can be chosen in such a way that |Ω| = |FM |.
Let O(λ) and O(µ) be two different W -orbits for integral strictly dominant elements λ and µ.
We say that the group Tm separates O(λ) and O(µ) if for any two elements λ1 ∈ O(λ) and
µ1 ∈ O(µ) there exists an element x ∈ Tm such that exp(2πi〈λ1, x〉) 6= exp(2πi〈µ1, x〉) (we use
here orbits, not signed orbits, since signs of points are not important for this reasoning). Note
that λ may coincide with µ.
60 A. Klimyk and J. Patera
Let f1 and f2 be two functions on En which are finite linear combinations of orbit functions.
We introduce a Tm-scalar product for them by the formula
〈f1, f2〉Tm =
∑
x∈Tm
f1(x)f2(x).
Then the following proposition is true (see [21] and [31]):
Proposition 17. If Tm separates the orbits O(λ) and O(µ), λ, µ ∈ P+
+ , then
〈ϕλ, ϕµ〉Tm = mn|W |δλµ. (11.9)
Proof. We have
〈ϕλ, ϕµ〉Tm =
∑
x∈Tm
∑
w∈W
∑
w′∈W
(detw)(detw′) exp(2πi〈wλ− w′µ, x〉)
=
∑
w∈W
∑
w′∈W
(detww′)
(∑
x∈Tm
exp(2πi〈wλ− w′µ, x〉)
)
.
Since Tm separates O(λ) and O(µ), then none of the differences wλ−w′µ in the last sum vanishes
on Tm. Since Tm is a group, one has∑
x∈Tm
exp(2πi〈wλ− w′µ, x〉) = mnδwλ,w′µ.
Therefore, 〈ϕλ, ϕµ〉Tm = mn|W |δλµ. Proposition is proved. �
Let f be an anti-invariant (with respect to W aff) function on En which is a finite linear
combination of antisymmetric orbit functions:
f(x) =
∑
λj∈P+
+
aλj
ϕλj
(x). (11.10)
Our aim is to determine f(x) by its values on a finite subset of En, namely, on Tm.
We suppose that Tm separate orbits O(λj) with λj from the right hand side of (11.10). Then
taking the Tm-scalar product of both sides of (11.10) with ϕλj
and using the relation (11.9) we
obtain
aλj
= (mn|W |)−1 〈f, ϕλj
〉Tm .
Let now s(1), s(2), . . . , s(h) be all elements of F ∩ 1
mQ
∨, which do not lie on some wall of Weyl
chambers. Then
aλj
= m−n|W |−1
∑
x∈Tm
f(x)ϕλj
(x) = mn
h∑
i=1
f(s(i))ϕλj
(s(i)). (11.11)
Thus, the finite number of values f(s(i)), i = 1, 2, . . . , h, of the function f(x) determines the
coefficients aλj
and, therefore, determines the function f(x) on the whole space El.
This means that we can reconstruct aW aff -anti-invariant function f(x) on the whole space En
by its values on the finite set FM under an appropriate value of M . Namely, we have to
expand this function, taken on FM , into the series (11.10) by means of the coefficients aλj
,
determined by formula (11.11), and then to continue analytically the expansion (11.10) to the
Antisymmetric Orbit Functions 61
whole fundamental domain F (and, therefore, to the whole space En), that is, to consider the
decomposition (11.10) for all x ∈ En.
We have assumed that the function f(x) is a finite linear combination of orbit functions.
If f(x) expands into infinite sum of orbit functions, then for applying the above procedure we
have to approximate the function f(x) by taking a finite number of terms in this infinite sum
and then apply the procedure. That is, in this case we obtain an approximate expression of the
function f(x) by using a finite number of its values.
At last, we explain how to choose a set Ω in formula (11.8). The set FM consists of the
points (11.6). This set determines the set of points
λ = s1ω1 + s2ω2 + · · ·+ snωn,
where s1, s2, . . . , sn run over the same values as for the set FM . The subset of this set, consisting
of strictly dominant elements, can be taken as the set Ω (see [31]).
11.5 Antisymmetric and symmetric multivariate discrete Fourier transforms
The discrete Fourier transform of Subsection 11.1 can be generalized to the n-dimensional case
in a symmetric or antisymmetric form without using the results of Subsection 11.4.
We take the discrete exponential function (11.1),
em(s) := N−1/2 exp(2πims), s ∈ FN ≡
{
1
N ,
2
N , . . . ,
N−1
N , 1
}
, m ∈ Z≥0,
and make a multivariate discrete exponential function taking a product of n copies of func-
tions (11.1):
EXPm(s) := em1(s1)em2(s2) · · · emn(sn) (11.12)
= N−n/2 exp(2πim1s1) exp(2πim2s2) · · · exp(2πimnsn), sj ∈ FN , mi ∈ Z≥0,
where s = (s1, s2, . . . , sn) and m = (m1,m2, . . . ,mn). Now we take these multivariate functions
for integers mi such that m1 > m2 > · · · > mn ≥ 0 and make an antisymmetrization. As
a result, we obtain a finite version of the antisymmetric orbit function (4.14):
em(s) := |Sn|−1/2 det(emi(sj))
n
i,j=1, (11.13)
where |Sn| is the order of the symmetric group Sn.
The n-tuples s in (11.13) runs over FnN ≡ FN × · · · × FN (n times). We denote by F̂nN the
subset of FnN consisting of s ∈ FnN such that
s1 > s2 > · · · > sn.
Note that acting by the permutations w ∈ Sn upon F̂nN we obtain the whole set FnN without
those points which are invariant under some nontrivial permutation w ∈ Sn. Clearly, the
function (11.13) vanishes on the last points.
Since the discrete exponential functions em(s) satisfy the equality em(s) = em+N (s), we do not
need to consider them for all valuesm ∈ Z≥0. It is enough to consider them form ∈ {1, 2, . . . , N}.
By D+
N we denote the set of integer n-tuples m = (m1,m2, . . . ,mn) such that
N ≥ m1 > m2 > · · · > mn > 0.
We need a scalar product in the space of linear combinations of the functions (11.12). It can
be given by the formula
〈EXPm(s),EXPm′(s)〉 ≡
n∏
i=1
〈emi(si), em′
i
(si)〉 :=
n∏
i=1
∑
si∈FN
emi(si)em′
i
(si) = δmm′ , (11.14)
mi,m
′
i ∈ {1, 2, . . . , N},
where we used the relation (11.2).
62 A. Klimyk and J. Patera
Proposition 18. For m,m′ ∈ D+
N the discrete functions (11.13) satisfy the orthogonality rela-
tion
〈em(s), em′(s)〉 = |Sn|
∑
s∈F̂n
N
em(s)em′(s) = δmm′ , (11.15)
where the scalar product is determined by formula (11.14).
Proof. Since m1 > m2 > · · · > mn > 0, then due to the definition of the scalar product we
have
〈em(s), em′(s)〉 =
∑
s∈Fn
N
em(s)em′(s)
= |Sn|−1
∑
w∈Sn
n∏
i=1
∑
si∈FN
emw(i)
(si)em′
w(i)
(si) = δmm′ , (11.16)
where (mw(1),mw(2), . . . ,mw(n)) is obtained from (m1,m2, . . . ,mn) by action by the permutation
w ∈ Sn. Since functions em(s) are antisymmetric with respect to Sn, then∑
s∈Fn
N
em(s)em′(s) = |Sn|
∑
s∈F̂n
N
em(s)em′(s),
where we have taken into account that em(s) vanishes on those s ∈ FnN for which there exists
w ∈ Sn, w 6= 1, such that ws = s. This proves the proposition. �
Let f be a function on F̂nN (or an antisymmetric function on FnN ). Then it can be expanded
in the functions (11.13) as
f(s) =
∑
m∈D+
N
amem(s). (11.17)
The coefficients am are determined by the formula
am = |Sn|
∑
m∈F̂n
N
f(s)em(s). (11.18)
The expansions (11.17) and (11.18) follow from the facts that numbers of elements in D+
N and
in F̂nN are the same, and that the matrix
(em(s))m∈D+
N ,s∈F̂
n
N
is unitary. We call expansions (11.17) and (11.18) antisymmetric multivariate discrete Fourier
transforms.
Let us also give a symmetric multivariate discrete Fourier transforms. For this we take the
multivariate exponential functions (11.12) for integers mi such that
N ≥ m1 ≥ m2 ≥ · · · ≥ mn ≥ 1
and make a symmetrization. We obtain a finite version of the symmetric orbit function (6.11)
in [1] for the case An:
Em(s) := |Sn|−1/2det+(emi(sj))
n
i,j=1 := |Sn|−1/2
∑
w∈Sn
n∏
i=1
emw(i)
(si). (11.19)
Antisymmetric Orbit Functions 63
The n-tuples s in (11.19) run over FnN ≡ FN × · · · × FN (n times). We denote by F̆nN the
subset of FnN consisting of s ∈ FnN such that
s1 ≥ s2 ≥ · · · ≥ sn.
Note that acting by the permutations w ∈ Sn upon F̆nN we obtain the whole set FnN , where each
point, having some coordinates mi coinciding, is repeated several times. Namely, a point s is
contained |Ss| times in {wF̆nN ;w ∈ Sn}, where Ss is the subgroup of Sn consisting of elements
w ∈ Sn such that ws = s. The number |Ss| is called a multiplicity of the point s in the set
{wF̆nN ;w ∈ Sn}.
By D̆+
N we denote the set of integer n-tuples m = (m1,m2, . . . ,mn) such that
N ≥ m1 ≥ m2 ≥ · · · ≥ mn ≥ 1.
Proposition 19. For m,m′ ∈ D̆+
N the discrete functions (11.19) satisfy the orthogonality rela-
tion
〈Em(s), Em′(s)〉 = |Sn|
∑
s∈F̆n
M
|Ss|−1Em(s)Em′(s) = |Sm|δmm′ . (11.20)
Proof. This proposition is proved in the same way as Proposition 18, but we have to take into
account a difference between F̆nM and F̂nM . Due to the definition of the scalar product we have
〈Em(s), Em′(s)〉 =
∑
s∈Fn
N
Em(s)Em′(s)
= |Sn|−1|Sm|
∑
w∈Sn
n∏
i=1
∑
si∈FN
emw(i)
(si)em′
w(i)
(si)
= |Sm|δmm′ ,
where (mw(1),mw(2), . . . ,mw(n)) is obtained from (m1,m2, . . . ,mn) by action by the permutation
w ∈ Sn. Here we have taken into account that additional summands appear (with respect to
(11.16)) because some summands on the right hand side of (11.19) may coincide.
Since functions Em(s) are symmetric with respect to Sn, then∑
s∈Fn
N
Em(s)Em′(s) = |Sn|
∑
s∈F̆n
N
|Ss|−1Em(s)Em′(s),
where we have taken into account that under action by Sn upon F̆nN a point s appears |Ss| times
in FnN . This proves the proposition. �
Let f be a function on F̆nN (or a symmetric function on FnN ). Then it can be expanded in
functions (11.19) as
f(s) =
∑
m∈D̆+
N
amEm(s). (11.21)
The coefficients am are determined by the formula
am = |Sn||Sm|−1
∑
m∈F̆n
N
|Ss|−1f(s)Em(s). (11.22)
The expansions (11.21) and (11.22) follow from the facts that numbers of elements in D̆+
N
and in F̆nN are the same and from the orhogonality relation (11.20). We call expansions (11.21)
and (11.22) symmetric multivariate discrete Fourier transforms.
64 A. Klimyk and J. Patera
11.6 Discrete sine and cosine transforms
The grid FM for A1 is of the form
FM (A1) =
{
0, 1
M ,
2
M , . . . ,
M−1
M , 1
}
. (11.23)
The points 0 and 1 belong to the boundary of the fundamental domain F (A1) of W aff(A1).
Therefore, antisymmetric orbit functions of A1 vanish on these points and
F−
M (A1) =
{
1
M ,
2
M , . . . ,
M−1
M
}
(M − 1 points)
(see Subsection 11.2). Since the antisymmetric orbit functions for A1 are of the form ϕλ(x) =
2i sin(πmθ) (see Example of Subsection 4.1), these functions on the grid FM are given by
ϕm(s) = 2i sin(πms), s ∈ FM , m ∈ Z≥. (11.24)
Since ϕm(s) = ϕm+M (s), we consider these discrete functions only for m ∈ DM := {1, 2, . . . ,
M − 1}. The orthogonality relation for these functions is of the form
〈ϕm, ϕm′〉 =
∑
s∈F−M
ϕm(s)ϕm′(s) = 2Mδmm′ , m,m′ ∈ DM
(see [22]). They determine the following expansion of functions, given on the grid F−
M :
f(s) =
M−1∑
m=1
amϕm(s), (11.25)
where the coefficients am are given by
am =
1
2M
∑
s∈F−M
f(s)ϕm(s). (11.26)
Formulas (11.25) and (11.26) determine the discrete sine transform.
The symmetric orbit functions for A1 are of the form φλ(x) = 2 cos(πmθ). Then these
functions on the grid FM (A1) are
φm(s) = 2 cos(πms), s ∈ FM , m ∈ {0, 1, 2, . . . ,M}. (11.27)
The scalar product of these functions is given by
〈φm, φm′〉 =
∑
s∈FM
csφm(s)φm′(s) = rmMδmm′ , (11.28)
where rm = 4 for m = 0,M and rm = 2 otherwise, cs = 1/2 for s = 0, 1 and cs = 1 otherwise.
The functions φm, given by (11.27), determine an expansion of functions on the grid FM as
f(s) =
M∑
m=0
bmφm(s), s ∈ FM , (11.29)
where the coefficients bm are given by
bm = r−1
m
∑
s∈FM
csf(s)φm(s). (11.30)
Formulas (11.29) and (11.30) determine the discrete cosine transform.
Antisymmetric Orbit Functions 65
11.7 Antisymmetric multivariate discrete sine transforms
The discrete sine and cosine transforms of the previous subsection can be generalized to the
n-dimensional case in symmetric or antisymmetric form. In fact, these generalizations are finite
antisymmetric orbit function transforms (11.10) and (11.11) for multivariate sine transforms
and finite symmetric orbit function transforms (9.11) and (9.12) in [1] for multivariate cosine
transforms. However, we give in this subsection a derivation of these multivariate sine and cosine
transforms, independent of the previous consideration. We need only the 1-dimensional discrete
sine and cosine transforms of the previous subsection.
We take the discrete sine function (11.24) and make a multivariate discrete sine function by
multiplying n copies of functions (11.24):
SINm(s) := (2i)n sin(πm1s1) sin(πm2s2) · · · sin(πmnsn), (11.31)
sj ∈ FM ≡ FM (A1), mi ∈ DM ≡ {1, 2, . . . , N − 1},
where s = (s1, s2, . . . , sn) and m = (m1,m2, . . . ,mn). Now we take these multivariate functions
for integers mi such that M > m1 > m2 > · · · > mn > 0 and make antisymmetrization. As a
result, we obtain a finite version of the orbit function (4.19):
ϕm(s) := (2i)n|Sn|−1/2 det(sinπmisj)ni,j=1, (11.32)
where |Sn| is the order of the symmetric group |Sn|. (We have here expressions sinπmisj , not
sin 2πmisj as in (4.19). Note that in (4.17)mi, i = 1, 2, . . . , n, run over integers and half-integers,
whereas in (11.32) mi run over integers. Thus, in fact we have replaced 2mi with half-integer
values of mi by mi with integer values of mi.)
The n-tuple s in (11.32) runs over F−
M (A1)n ≡ F−
M (A1)×· · ·×F−
M (A1) (n times). We denote
by F̂nM the subset of F−
M (A1)n consisting of s ∈ F−
M (A1)n such that
s1 > s2 > · · · > sn.
Note that si here may take the values 1
M ,
2
M , . . . ,
M−1
M . Acting by permutations w ∈ Sn upon F̂nM
we obtain the whole set F−
M (A1)n without those points which are invariant under some nontrivial
permutation w ∈ Sn. Clearly, the function (11.32) vanishes on the last points.
We denote by D+
M the set of integer n-tuples m = (m1,m2, . . . ,mn) such that
M > m1 > m2 > · · · > mn > 0.
We wish to have a scalar product of functions (11.32). For this we define a scalar product of
functions (11.31) as
〈SINm(s),SINm′(s)〉 =
n∏
i=1
〈ϕmi(si), ϕm′
i
(si)〉,
where the scalar product 〈ϕmi(si), ϕm′
i
(si)〉 is given in Subsection 11.6. Since functions ϕm(s)
are linear combinations of functions SINm′(s), a scalar product for ϕm(s) is also defined.
Proposition 20. For m,m′ ∈ D+
M , the discrete functions (11.32) satisfy the orthogonality
relation
〈ϕm(s), ϕm′(s)〉 :=
∑
s∈F−M (A1)n
ϕm(s)ϕm′(s) = |Sn|
∑
s∈F̂n
M
ϕm(s)ϕm′(s) = (2M)nδmm′ . (11.33)
66 A. Klimyk and J. Patera
Proof. Since M > m1 > m2 > · · · > mn > 0, then due to the orthogonality relation for the
sine functions 2i sin(πms) (see the previous subsection) we have
∑
s∈F−M (A1)n
ϕm(s)ϕm′(s) = 4n|Sn|−1
∑
w∈Sn
n∏
i=1
M−1∑
si=1
sin(πmw(i)si) sin(πm′
w(i)si) = (2M)nδmm′ ,
where (mw(1),mw(2), . . . ,mw(n)) is obtained from (m1,m2, . . . ,mn) by action by the permutation
w ∈ Sn. Since functions ϕm(s) are antisymmetric with respect to Sn, we have∑
s∈F−M (A1)n
ϕm(s)ϕm′(s) = |Sn|
∑
s∈F̂n
M
ϕm(s)ϕm′(s).
This proves the proposition. �
Let f be a function on F̂nM (or an antisymmetric function on F−
M (A1)n). Then it can be
expanded in functions (11.32) as
f(s) =
∑
m∈D+
M
amϕm(s), (11.34)
where the coefficients am are determined by the formula
am = (2M)−n|Sn|
∑
m∈F̂n
M
f(s)ϕm(s). (11.35)
A validity of the expansions (11.34) and (11.35) follows from the facts that numbers of
elements in D+
M and in F̂nM are the same and from the orthogonality relation (11.33).
11.8 Symmetric multivariate discrete cosine transforms
We take the discrete cosine functions (11.27) and make multivariate discrete cosine functions by
multiplying n copies of these functions:
COSm(s) := φm1(s1)φm2(s2) · · ·φmn(sn) = 2n cos(πm1s1) cos(πm2s2) · · · cos(πmnsn),
sj ∈ FM ≡ FM (A1), mi ∈ {0, 1, 2, . . . ,M}. (11.36)
We take these functions for integers mi such that M ≥ m1 ≥ m2 ≥ · · · ≥ mn ≥ 0 and make
a symmetrization. As a result, we obtain a finite version of the orbit function (4.24):
φm(s) := 2n|Sn|−1/2
∑
w∈Sn
cosπmw(1)s1 · cosπmw(2)s2 · · · cosπmw(n)sn. (11.37)
(We have here expressions cosπmisj , not cos 2πmisj as in (4.24).)
The n-tuple s in (11.37) runs over FnM ≡ FM (A1)n. We denote by F̆nM the subset of FnM
consisting of s ∈ FnM such that
s1 ≥ s2 ≥ · · · ≥ sn.
Note that si here may take the values 0, 1
M ,
2
M , . . . ,
M−1
M , 1. Acting by permutations w ∈ Sn
upon F̆nM we obtain the whole set FnM , where points, invariant under some nontrivial permutation
w ∈ Sn, are repeated several times. It is easy to see that a point s0 ∈ FnM is repeated |Ss0 | times
in the set {wF̆nM ; w ∈ Sn}, where |Ss0 | is an order of the subgroup Ss0 ⊂ Sn, whose elements
leaves s0 invariant.
Antisymmetric Orbit Functions 67
We denote by D̆+
M the set of integer n-tuples m = (m1,m2, . . . ,mn) such that
M ≥ m1 ≥ m2 ≥ · · · ≥ mn ≥ 0.
A scalar product of functions (11.36) is determined by
〈COSm(s),COSm′(s)〉 =
n∏
i=1
〈φmi(si), φm′
i
(si)〉,
where the scalar product 〈φmi(si), φm′
i
(si)〉 is given by (11.28). Since functions φm(s) are linear
combinations of functions COSm′(s), then a scalar product for φm(s) is also defined.
Proposition 21. For m,m′ ∈ D̆+
M , the discrete functions (11.37) satisfy the orthogonality
relation
〈φm(s), φm′(s)〉 =
∑
s∈Fn
M
csφm(s)φm′(s) = |Sn|
∑
s∈F̆n
M
|Ss|−1csφm(s)φm′(s)
= Mnrm|Sm|δmm′ , (11.38)
where cs = cs1cs2 · · · csn, rs = rm1rm2 · · · rmn, and csi and rmi are such as in (11.28).
Proof. Due to the orthogonality relation for the cosine functions φm(s) = 2 cos(πms) (see
formula (11.28)) we have
∑
s∈Fn
M
csφm(s)φm′(s) = 4n|Sn|−1|Sm|
∑
w∈Sn
n∏
i=1
M∑
si=0
csi cos(πmw(i)si) cos(πm′
w(i)si)
= |Sm|Mnrmδmm′ , (11.39)
where (mw(1),mw(2), . . . ,mw(n)) is obtained from (m1,m2, . . . ,mn) by action by the permutation
w ∈ Sn. Since functions φm(s) are symmetric with respect to Sn, we have∑
s∈FM (A1)n
csφm(s)φm′(s) = |Sn|
∑
s∈F̆n
M
|Ss|−1csφm(s)φm′(s).
This proves the proposition. �
Let f be a function on F̆nM (or an antisymmetric function on FnM ). Then it can be expanded
in functions (11.37) as
f(s) =
∑
m∈D̆+
M
amφm(s), (11.40)
where the coefficients am are determined by the formula
am = M−n|Sm|−1r−1
m 〈f(s), φm(s)〉
= M−n|Sm|−1r−1
m |Sn|
∑
s∈F̆n
M
|Ss|−1csf(s)φm(s). (11.41)
A validity of the expansions (11.40) and (11.41) follows from the fact that numbers of elements
in D̆+
M and F̆nM are the same and from the orthogonality relation (11.38).
68 A. Klimyk and J. Patera
11.9 Other discrete cosine transforms
Along with the discrete cosine transform of Subsection 11.6 there are other discrete transforms
with the discrete cosine function as a kernel. In [34] the discrete cosine transforms are called as
DCT-1, DCT-2, DCT-3, DCT-4. The transform DCT-1 is in fact the transform, considered in
Subsection 11.6. Let us describe all these transforms (including the transform DCT-1). They
are determined by a positive integer N .
DCT-1. This transform is given by the kernel
µr(k) = cos
πrk
N
(11.42)
(we preserve the notation used in the literature on signal processing), where
k, r ∈ {0, 1, 2, . . . , N}. (11.43)
The orthogonality relation for these discrete functions is given by
N∑
k=0
ck cos
πrk
N
cos
πr′k
N
= hr
N
2
δrr′ , (11.44)
where ck = 1/2 for k = 0, N and ck = 1 otherwise, hr = 2 for r = 0, N and hr = 1 otherwise.
Thus, these functions give the expansion
f(k) =
N∑
r=0
ar cos
πrk
N
, where ar =
2
hrN
N∑
k=0
ckf(k) cos
πrk
N
. (11.45)
DCT-2. This transform is given by the kernel
ωr(k) = cos
π(r + 1
2)k
N
, (11.46)
where
k, r ∈ {0, 1, 2, . . . , N − 1}.
The orthogonality relation for these discrete functions is given by
N−1∑
k=0
ck cos
π(r + 1
2)k
N
cos
π(r′ + 1
2)k
N
=
N
2
δrr′ , (11.47)
where ck = 1/2 for k = 0 and ck = 1 otherwise.
These functions determine the expansion
f(k) =
N−1∑
r=0
arωr(k), where ar =
2
N
N−1∑
k=0
ckf(k)ωr(k). (11.48)
DCT-3. This transform is determined by the kernel
σr(k) = cos
πr(k + 1
2)
N
, (11.49)
Antisymmetric Orbit Functions 69
where k and r run over the values {0, 1, 2, . . . , N − 1}. The orthogonality relation for these
discrete functions is given by the formula
N−1∑
k=0
cos
πr(k + 1
2)
N
cos
πr′(k + 1
2)
N
= hr
N
2
δrr′ , (11.50)
where hk = 2 for k = 0 and hk = 1 otherwise.
These functions give the expansion
f(k) =
N−1∑
r=0
ar cos
πr(k + 1
2)
N
, where ar =
2
hrN
N−1∑
k=0
f(k) cos
πr(k + 1
2)
N
. (11.51)
DCT-4. This transform is given by the kernel
τr(k) = cos
π(r + 1
2)(k + 1
2)
N
, (11.52)
where k and r run over the values {0, 1, 2, . . . , N − 1}. The orthogonality relation for these
discrete functions is given by
N−1∑
k=0
cos
π(r + 1
2)(k + 1
2)
N
cos
π(r′ + 1
2)(k + 1
2)
N
=
N
2
δrr′ . (11.53)
These functions determine the expansion
f(k) =
N−1∑
r=0
ar cos
π(r + 1
2)(k + 1
2)
N
, (11.54)
where
ar =
2
N
N−1∑
k=0
f(k) cos
π(r + 1
2)(k + 1
2)
N
.
Note that there also exist four discrete sine transforms, corresponding to the above discrete
cosine transforms. They are obtained from the cosine transforms by replacing in (11.45), (11.48),
(11.51) and (11.54) cosines discrete functions by sine discrete functions (see [12] and [35]).
11.10 Other antisymmetric multivariate discrete cosine transforms
Each of the discrete cosine transforms DCT-1, DCT-2, DCT-3, DCT-4 generates the corres-
ponding antisymmetric multivariate discrete cosine transforms. We call these transforms as
AMDCT-1, AMDCT-2, AMDCT-3 and AMDCT-4. Let us give these transforms without proof.
Their proofs are the same as in the case of antisymmetric multivariate discrete cosine transforms
of Subsection 11.7. Below we use the notation Dn,−
N for the subset of the set Dn
N ≡ DN ×DN ×
· · · ×DN (n times) with DN = {0, 1, 2, . . . , N} consisting of points r = (r1, r2, . . . , rn), ri ∈ DN ,
such that
N ≥ r1 > r2 > · · · > rn ≥ 0.
AMDCT-1. This transform is given by the kernel
Mr(k) = |Sn|−1/2 det (µri(kj))
n
i,j=1 , (11.55)
70 A. Klimyk and J. Patera
where µr(k) = cos πrkN and k = (k1, k2, . . . , kn), ki ∈ {0, 1, 2, . . . , N}. The orthogonality relation
for these kernels is
〈Mr(k),Mr′(k)〉 = |Sn|
∑
k∈Dn,−
N
ckMr(k)Mr′(k) = hr
(
N
2
)n
δrr′ , (11.56)
where
ck = c1c2 · · · cn, hk = h1h2 · · ·hn,
and ci and hj are such as in formula (11.44).
This transform is given by the formula
f(k) =
∑
r∈Dn,−
N
arMr(k), where ar = h−1
r |Sn|
(
2
N
)n ∑
k∈Dn,−
N
ckf(k)Mr(k). (11.57)
The Plancherel formula for this transform is
|Sn|
∑
k∈Dn,−
N
ck|f(k)|2 =
(
N
2
)n ∑
r∈Dn,−
N
hr|ar|2.
AMDCT-2. We use the subset Dn,−
N−1 of the set Dn
N−1 with DN−1 = {0, 1, 2, . . . , N − 1}
consisting of points r = (r1, r2, . . . , rn), ri ∈ DN−1, such that
N − 1 ≥ r1 > r2 > · · · > rn ≥ 0.
This transform is given by the kernel
Ωr(k) = |Sn|−1/2 det (ωri(kj))
n
i,j=1 , (11.58)
where ωr(k) = cos π(r+ 1
2
)k
N and k = (k1, k2, . . . , kn), ki ∈ {0, 1, 2, . . . , N − 1}. The orthogonality
relation for these kernels is
〈Ωr(k),Ωr′(k)〉 = |Sn|
∑
k∈Dn,−
N−1
ckΩr(k)Ωr′(k) =
(
N
2
)n
δrr′ , (11.59)
where ck = c1c2 · · · cn and ci are such as in formula (11.47).
This transform is given by the formula
f(k) =
∑
r∈Dn,−
N−1
arΩr(k), where ar = |Sn|
(
2
N
)n ∑
k∈Dn,−
N−1
ckf(k)Ωr(k). (11.60)
The Plancherel formula for this transform is of the form
|Sn|
∑
k∈Dn,−
N−1
ck|f(k)|2 =
(
N
2
)n ∑
r∈Dn,−
N−1
|ar|2.
AMDCT-3. This transform is given by the kernel
Σr(k) = |Sn|−1/2 det (σri(kj))
n
i,j=1 , r ∈ Dn,−
N−1, kj ∈ DN−1, (11.61)
Antisymmetric Orbit Functions 71
where σr(k) = cos πr(k+
1
2
)
N . The orthogonality relation for these kernels is
〈Σr(k),Σr′(k)〉 = |Sn|
∑
k∈Dn,−
N−1
Σr(k)Σr′(k) = hr
(
N
2
)n
δrr′ , (11.62)
where hr = h1h2 · · ·hn and hj are such as in formula (11.50).
This transform is given by the formula
f(k) =
∑
r∈Dn,−
N−1
arΣr(k), where ar = h−1
r |Sn|
(
2
N
)n ∑
k∈Dn,−
N−1
f(k)Σr(k). (11.63)
The Plancherel formula is of the form
|Sn|
∑
k∈Dn,−
N−1
|f(k)|2 =
(
N
2
)n ∑
r∈Dn,−
N−1
hr|ar|2.
AMDCT-4. This transform is given by the kernel
Tr(k) = |Sn|−1/2 det (τri(kj))
n
i,j=1 , r ∈ Dn,−
N−1, kj ∈ DN−1, (11.64)
where τr(k) = cos π(k+ 1
2
)(r+ 1
2
)
N . The orthogonality relation for these kernels is
〈Tr(k), Tr′(k)〉 = |Sn|
∑
k∈Dn,−
N−1
Tr(k)Tr′(k) =
(
N
2
)n
δrr′ . (11.65)
This transform is given by the formula
f(k) =
∑
r∈Dn,−
N−1
arTr(k), where ar = |Sn|
(
2
N
)n ∑
k∈Dn,−
N−1
f(k)Tr(k). (11.66)
The Plancherel formula for this transform is
|Sn|
∑
k∈Dn,−
N−1
|f(k)|2 =
(
N
2
)n ∑
r∈Dn,−
N−1
|ar|2.
11.11 Other symmetric multivariate discrete cosine transforms
To each of the discrete cosine transforms DCT-1, DCT-2, DCT-3, DCT-4 there corresponds
a symmetric multivariate discrete cosine transform. We denote the corresponding transforms as
SMDCT-1, SMDCT-2, SMDCT-3, SMDCT-4. Below we give these transforms without proof
(proofs are the same as in the case of symmetric multivariate discrete cosine transforms of
Subsection 11.8). We fix a positive integer N and use the notation Dn,+
N for the subset of
the set Dn
N ≡ DN ×DN × · · · ×DN (n times) with DN = {0, 1, 2, . . . , N} consisting of points
r = (r1, r2, . . . , rn), ri ∈ Z≥0 such that
N ≥ r1 ≥ r2 ≥ · · · ≥ rn ≥ 0.
The set Dn,+
N is an extension of the set Dn,−
N from the previous subsection by adding points
which are invariant with respect of some elements of the permutation group Sn.
72 A. Klimyk and J. Patera
The set Dn
N is obtained by action by elements of the group Sn upon Dn,+
N , that is, Dn
N coin-
cides with the set {wDn,+
N ;w ∈ Sn}. However, in {wDn,+
N ;w ∈ Sn}, some points occur several
times. Namely, a point k0 ∈ Dn,+
N occurs |Sk0 | times in the set {wDn,+
N ;w ∈ Sn}, where |Sk0 | is
an order of the subgroup Sk0 ⊂ Sn consisting of elements w ∈ Sn leaving k0 invariant.
SMDCT-1. This transform is given by the kernel
M̂r(k) = |Sn|−1/2
∑
w∈Sn
µrw(1)
(k1)µrw(2)
(k2) · · ·µrw(n)
(kn), (11.67)
where, as before, µr(k) = cos πrkN is the discrete cosine function from Subsection 11.9, k =
(k1, k2, . . . , kn), ki ∈ {0, 1, 2, . . . , N}, and the set (w(1), w(2), . . . , w(n)) is obtained from the set
(1, 2, . . . , n) by applying the permutation w ∈ Sn. The orthogonality relation for these kernels
is
〈M̂r(k), M̂r′(k)〉 = |Sn|
∑
k∈Dn,+
N
|Sk|−1ckM̂r(k)M̂r′(k) = hr
(
N
2
)n
|Sr|δrr′ , (11.68)
where Sr is the subgroup of Sn consisting of elements leaving r invariant,
ck = c1c2 · · · cn, hk = h1h2 · · ·hn,
and ci and hj are such as in formula (11.44).
This transform is given by the formula
f(k) =
∑
r∈Dn,+
N
arM̂r(k), (11.69)
where
ar = h−1
r |Sr|−1|Sn|
(
2
N
)n ∑
k∈Dn,+
N
|Sk|−1ckf(k)M̂r(k).
The Plancherel formula for this transform is
|Sn|
∑
k∈Dn,+
N
|Sk|−1ck|f(k)|2 =
(
N
2
)n ∑
r∈Dn,+
N
hr|Sr||ar|2.
This transform is in fact a variation of the symmetric multivariate discrete cosine transforms
from Subsection 11.8.
SMDCT-2. This transform is given by the kernel
Ω̂r(k) = |Sn|−1/2
∑
w∈Sn
ωrw(1)
(k1)ωrw(2)
(k2) · · ·ωrw(n)
(kn), (11.70)
r ∈ Dn,+
N−1, rj ∈ DN−1 ≡ {0, 1, 2, . . . , N − 1},
where ωr(k) = cos π(r+ 1
2
)k
N and k = (k1, k2, . . . , kn), ki ∈ {0, 1, 2, . . . , N − 1}. The orthogonality
relation for these kernels is
〈Ω̂r(k), Ω̂r′(k)〉 = |Sn|
∑
k∈Dn,+
N−1
|Sk|−1ckΩ̂r(k)Ω̂r′(k) =
(
N
2
)n
|Sr|δrr′ , (11.71)
Antisymmetric Orbit Functions 73
where Dn,+
N−1 is the set Dn,+
N with N replaced by N − 1, ck = c1c2 · · · cn and cj are such as
in (11.47).
This transform is given by the formula
f(k) =
∑
r∈Dn,+
N−1
arΩ̂r(k), (11.72)
where
ar = |Sn||Sr|−1
(
2
N
)n ∑
k∈Dn,+
N−1
|Sk|−1ckf(k)Ω̂r(k).
The Plancherel formula for this transform is of the form
|Sn|
∑
k∈Dn,+
N−1
|Sk|−1ck|f(k)|2 =
(
N
2
)n ∑
r∈Dn,+
N−1
|Sr||ar|2.
SMDCT-3. This transform is given by the kernel
Σ̂r(k) = |Sn|−1/2
∑
w∈Sn
σrw(1)
(k1)σrw(2)
(k2) · · ·σrw(n)
(kn), (11.73)
r ∈ Dn,+
N−1, rj ∈ DN−1,
where σr(k) = cos πr(k+
1
2
)
N . The orthogonality relation for these kernels is
〈Σ̂r(k), Σ̂r′(k)〉 = |Sn|
∑
k∈Dn,+
N−1
|Sk|−1Σ̂r(k)Σ̂r′(k) = hr
(
N
2
)n
|Sr|δrr′ , (11.74)
where hr = h1h2 · · ·hn and hi are such as in formula (11.50).
This transform is given by the formula
f(k) =
∑
r∈Dn,+
N−1
arΣ̂r(k), (11.75)
where
ar = h−1
r |Sr|−1|Sn|
(
2
N
)n ∑
k∈Dn,+
N−1
|Sk|−1f(k)Σ̂r(k).
The Plancherel formula is of the form
|Sn|
∑
k∈Dn,+
N−1
|Sk|−1|f(k)|2 =
(
N
2
)n ∑
r∈Dn,+
N−1
hr|Sr||ar|2.
SMDCT-4. This transform is given by the kernel
T̂r(k) = |Sn|−1/2
∑
w∈Sn
τrw(1)
(k1)τrw(2)
(k2) · · · τrw(n)
(kn), (11.76)
r ∈ Dn,+
N−1, rj ∈ DN−1,
74 A. Klimyk and J. Patera
where τr(k) = cos π(k+ 1
2
)(r+ 1
2
)
N . The orthogonality relation for these kernels is
〈T̂r(k), T̂r′(k)〉 = |Sn|
∑
k∈Dn,+
N−1
|Sk|−1T̂r(k)T̂r′(k) =
(
N
2
)n
|Sr|δrr′ . (11.77)
This transform is given by the formula
f(k) =
∑
r∈Dn,+
N−1
arT̂r(k), (11.78)
where
ar =
(
2
N
)n
|Sr|−1|Sn|
∑
k∈Dn,+
N−1
|Sk|−1f(k)T̂r(k).
The Plancherel formula for this transform is
|Sn|
∑
k∈Dn,+
N−1
|Sk|−1|f(k)|2 =
(
N
2
)n ∑
r∈Dn,+
N−1
|Sr||ar|2.
12 Solutions of the Laplace equation
on n-dimensional simplexes
We have seen in [1] that symmetric orbit functions are solutions of the Neumann boundary
value problem on n-dimensional simplexes. That is, they are solutions of the Laplace equation
∆f(x) = λf(x) on the fundamental domain F of the corresponding affine Weyl group W aff
satisfying the Neumann boundary condition
∂φλ(x)
∂m
∣∣∣∣
∂F
= 0 , λ ∈ P+, (12.1)
where ∂F is the (n − 1)-dimensional boundary of F and m is the normal to the boundary. In
this section we show that antisymmetric orbit functions are solutions of the Laplace operator,
which vanish on the boundary ∂F of the fundamental domain F .
12.1 The case of n-dimensional simplexes related to An, Bn, Cn and Dn
Let F be the fundamental domain of one of the affine Weyl groups W aff(An), W aff(Bn),
W aff(Cn), W aff(Dn) (see Subsection 5.9 for an explicit form of these domains). We use orthogo-
nal coordinates x1, x2, . . . , xn+1 on F in the case of W aff(An) and the orthogonal coordinates
x1, x2, . . . , xn in other cases (see Section 3). Thus the fundamental domain F for W aff(An) is
placed in the hyperplane x1 + x2 + · · ·+ xn+1 = 0.
The Laplace operator on F in the orthogonal coordinates has the form
∆ =
∂2
∂x2
1
+
∂2
∂x2
2
+ · · ·+ ∂2
∂x2
r
,
where r = n + 1 for An and r = n for Bn, Cn and Dn. Let us consider the case Bn. We take
a summand from the expression (4.16) for the antisymmetric orbit function ϕλ(x) of Bn and act
upon it by the operator ∆. We get
∆e2πi((w(ελ))1x1+···+(w(ελ))nxn)
Antisymmetric Orbit Functions 75
= −4π2[(ε1m1)2 + · · ·+ (εnmn)2]e2πi((w(ελ))1x1+···+(w(ελ))nxn)
= −4π2(m2
1 + · · ·+m2
n)e
2πi((w(ελ))1x1+···+(w(ελ))nxn)
= −4π2〈λ, λ〉 e2πi((w(ελ))1x1+···+(w(ελ))nxn),
where λ = (m1,m2, . . . ,mn) is the weight, determining ϕλ(x), in the orthogonal coordinates
and w ∈ Sn. Since this action of ∆ does not depend on a summand from (4.16), we have
∆ϕλ(x) = −4π2〈λ, λ〉ϕλ(x). (12.2)
For An, Cn and Dn this formula also holds and the corresponding proofs are the same. Remark
that in the case An the scalar product 〈λ, λ〉 is equal to
〈λ, λ〉 = m2
1 +m2
2 + · · ·+m2
n+1.
The formula (12.2) can be generalized in the following way. Let σk(y1, y2, . . . , yr) be the k-th
elementary symmetric polynomial of degree k of the variables y1, y2, . . . , yr, that is,
σk(y1, y2, . . . , yr) =
∑
1≤k1<k2<···<kr≤n
yk1yk2 · · · ykr .
Then
σk
(
∂2
∂x2
1
, ∂
2
∂x2
2
, . . . , ∂
2
∂x2
r
)
ϕλ(x) = (−4π2)kσk(m2
1,m
2
2, . . . ,m
2
r)ϕλ(x), k = 1, 2, . . . , r, (12.3)
where, as before, r = n+ 1 for An and r = n for Bn, Cn and Dn. r differential equations (12.3)
are algebraically independent.
Thus, antisymmetric orbit functions are eigenfunctions of the operator σk
(
∂2
∂x2
1
, ∂
2
∂x2
2
, . . . , ∂
2
∂x2
r
)
,
k = 1, 2, . . . , n, on the fundamental domain F satisfying the boundary condition
ϕλ(x) = 0, λ ∈ D+, (12.4)
(see Subsection 5.3).
12.2 The Laplace operator in ω-basis
We may parametrize elements of F by coordinates in the ω-basis: x = θ1ω1+· · ·+θ2ω2. Denoting
by ∂k partial derivative with respect to θk, we have the Laplace operator ∆ in the form
∆ =
1
2
n∑
i,j=1
〈αj , αj〉−1Mij∂i∂j , (12.5)
where (Mij) is the corresponding Cartan matrix. One can see that it is indeed the Laplace
operator as follows. The matrix (Sij) = (〈αj , αj〉−1Mij) is symmetric with respect to transpo-
sition and its determinant is positive. Hence it can be diagonalized, so that ∆ becomes a sum
of second derivatives with no mixed derivative terms.
12.3 Rank two and three special cases
We write down explicit form of the Laplace operators in coordinates in the ω-basis for ranks 2
and 3. For rank two the operator ∆ is of the form
A2 : (∂2
1 − ∂1∂2 + ∂2
2)ϕ = −4π2
3 (a2 + ab+ b2)ϕ, F = {0, ω1, ω2}, (12.6)
76 A. Klimyk and J. Patera
C2 : (2∂2
1 − 2∂1∂2 + ∂2
2)ϕ = −2π2(a2 + 4ab+ 4b2)ϕ, F = {0, ω1, ω2}, (12.7)
G2 : (∂2
1 − 3∂1∂2 + 3∂2
2)ϕ = −4π2
3 (3a2 + 3ab+ b2)ϕ, F = {0, ω1
2 , ω2}. (12.8)
Here, to simplify notation, ϕ stands for ϕλ(x), λ = (a b) and x = (θ1 θ2). Although the
same symbols are used for analogous objects in the three cases, their geometric meaning is very
different. It is given by the appropriate Cartan matrix M from (2.1). In particular, the vertices
of F form an equilateral triangle in the case of A2, for C2 the triangle is half of a square, and it
is a half of an equilateral triangle for G2.
In the semisimple case A1 ×A1 one has M = 2 ( 1 0
0 1 ), therefore ∆ = 2∂2
1 + 2∂2
2 , and ϕλ(x) is
the product of two antisymmetric orbit functions, one from each A1. The fundamental domain
is a square.
There are three 3-dimensional cases to consider, namely A3, B3, and C3. In addition there
are four cases involving non-simple groups of the same rank. For A3, B3, and C3 the result can
be represented by the formulas
A3 : ∆ = ∂2
1 + ∂2
2 + ∂2
3 − ∂1∂2 − ∂2∂3,
B3 : ∆ = ∂2
1 + ∂2
2 + 2∂2
3 − ∂1∂2 − 2∂2∂3,
C3 : ∆ = 2∂2
1 + 2∂2
2 + 2∂2
3 − 2∂1∂2 − 2∂2∂3. (12.9)
12.4 Antisymmetric orbit functions as eigenfunctions of other operators
Antisymmetric orbit functions are eigenfunctions of many other operators. We consider examples
of such operators.
We associate with each y ∈ En the shift operator Ty which acts on the exponential functions
e2πi〈λ,x〉 as
Tye
2πi〈λ,x〉 = e2πi〈λ,x+y〉 = e2πi〈λ,y〉e2πi〈λ,x〉.
The action of elements of the Weyl group W on functions, given on En, is given as wf(x) =
f(wx). For each y ∈ En we define an operator acting on orbit functions by the formula
Dy =
∑
w∈W
(detw)wTy.
Then
Dyϕλ(x) = Dy
∑
w∈W
(detw)e2πi〈wλ,x〉
=
∑
w′∈W
(detw′)
∑
w∈W
(detw)e2πi〈wλ,y〉e2πi〈wλ,w′x〉
=
∑
w∈W
(detw)e2πi〈wλ,y〉
∑
w′∈W
(detw′)e2πi〈wλ,w′x〉
=
∑
w∈W
(detw)e2πi〈wλ,y〉
∑
w′∈W
(detw′)e2πi〈w′−1wλ,x〉
=
∑
w∈W
e2πi〈wλ,y〉
∑
w′∈W
(detw′−1
w)e2πi〈w′−1wλ,x〉
=
∑
w∈W
e2πi〈wλ,y〉ϕλ(x) = φλ(y)ϕλ(x),
that is, ϕλ(x) is an eigenfunction of the operator Dy with eigenvalue φλ(y).
Antisymmetric Orbit Functions 77
Now we consider the operator
D̂y =
∑
w∈W
wTy.
Then, conducting the same reasoning as above, we receive the relation
D̂yϕλ(x) = ϕλ(y)ϕλ(x),
that is, ϕλ(x) is an eigenfunction of the operator D̂y with eigenvalue ϕλ(y).
It is proved in the same way that
Dyφλ(x) = ϕλ(y)φλ(x),
that is, the symmetric orbit function φλ(x) is an eigenfunction of the operatorDy with eigenvalue
ϕλ(y).
It is shown similarly that in the cases of An, Bn, Cn, Dn antisymmetric orbit functions ϕλ(x)
are eigenfunctions of the operators
∑
w∈W
w
∂2
∂x2
i
, i = 1, 2, . . . , r,
where x1, x2, . . . , xr are orthogonal coordinates of the point x, r = n + 1 for An and r = n for
other cases. In fact, these operators are multiple to the Laplace operator ∆.
It is easy to show that in the cases of An, Bn, Cn, and also of Dn with even n, antisymmetric
orbit functions ϕλ(x) are solutions of the equations∑
w∈W
w
∂
∂xi
f = 0, i = 1, 2, . . . , r.
13 Symmetric and antisymmetric functions
Symmetric (antisymmetric) orbit functions are symmetrized (antisymmetrized) versions of the
exponential function, when symmetrization (antisymmetrization) is fulfilled by a Weyl group.
Instead of the exponential function we can take any other set of functions, for example, a set
of orthogonal polynomials or a countable set of functions. Then we obtain a corresponding set
of orthogonal symmetric (antisymmetric) polynomials or a set of symmetric (antisymmetric)
functions. Such sets of polynomials and functions are a subject of investigation in this section.
13.1 Symmetrization and antisymmetrization
by (anti)symmetric orbit functions
Symmetric and antisymmetric orbit functions can be used for symmetrization and antisym-
metrization of functions. Let um(x), m = 0, 1, 2, . . . , be a set of continuous functions of one
variables. We create functions of n variables
ui1,i2,...,in(x1, x2, . . . , xn) ≡ ui1(x1)ui2(x2) · · ·uin(xn), ik = 0, 1, 2, . . . .
Then the functions
ũi1,i2,...,in(λ1, λ2, . . . , λn) =
∫
F
ui1,i2,...,in(x1, x2, . . . , xn)φλ(x1, x2, . . . , xn)dx, (13.1)
78 A. Klimyk and J. Patera
where λ ≡ (λ1, λ2, . . . , λn), φλ(x) is a symmetric orbit function, and dx is the Euclidean measure
on En (that is, dx = dx1 · · · dxn), is symmetric with respect to the action of the Weyl group W .
Indeed, for w ∈W we have
ũi1,i2,...,in(wλ) =
∫
F
ui1,i2,...,in(x1, x2, . . . , xn)φwλ(x1, x2, . . . , xn)dx
=
∫
F
ui1,i2,...,in(x1, x2, . . . , xn)φλ(x1, x2, . . . , xn)dx = ũi1,i2,...,in(λ).
Similarly, the functions
ûi1,i2,...,in(λ1, λ2, . . . , λn) =
∫
F
ui1,i2,...,in(x1, x2, . . . , xn)ϕλ(x1, x2, . . . , xn)dx, (13.2)
where ϕλ(x) is an antisymmetric orbit function, are antisymmetric with respect to the action of
the Weyl group W . In particular, the functions ûi1,i2,...,in(λ) vanish on Weyl chambers.
Formulas (13.1) and (13.2) are used for obtaining symmetric and antisymmetric functions or
polynomials.
13.2 Eigenfunctions of (anti)symmetric orbit function transform for W (An)
Let Hn(x), n = 0, 1, 2, . . . , be the well-known Hermite polynomials. They are defined by the
formula
Hn(x) = n!
[n/2]∑
m=0
(−1)m(2x)n−2m
m!(n− 2m)!
where [n/2] is an integral part of the number n/2. These polynomials obey the difference
equation(
d2
dx2
− 2x
d
dx
+ 2n
)
Hn(x) = 0. (13.3)
They satisfy the relation
1√
2π
∫ ∞
−∞
eipxe−p
2/2Hm(p)dp = i−me−x
2/2Hm(x)
(see, for example, Subsection 12.2.4 in [36]), which can be written in the form∫ ∞
−∞
e2πipxe−πp
2
Hm(
√
2πp)dp = i−me−πx
2
Hm(
√
2πx). (13.4)
Using the Hermite polynomials we create polynomials of many variables
Hm(x) ≡ Hm1,m2,...,mn(x1, x2, . . . , xn) := Hm1(x1)Hm2(x2) · · ·Hmn(xn). (13.5)
The functions
e−|x|
2/2Hm(x), mi = 0, 1, 2, . . . , i = 1, 2, . . . , n, (13.6)
form an orthogonal basis of the Hilbert space L2(Rn) with the scalar product
〈f1, f2〉 :=
∫
Rn
f1(x)f2(x)dx,
where dx = dx1 dx2 · · · dxn.
Antisymmetric Orbit Functions 79
The polynomials Hm(x) satisfy the differential equation(
∆− 2
n∑
i=1
xi
∂
∂xi
+ 2|m|
)
Hm(x) = 0, (13.7)
where ∆ is the Laplace operator ∆ =
n∑
m=0
∂2
∂x2
m
and |m| = m1 +m2 + · · ·+mn.
We make symmetrization and antisymmetrization of the functions
Hm(x) := e−π|x|Hm(
√
2πx)
(obtained from (13.6) by replacing x by
√
2πx) by means of orbit functions of An−1:∫
Rn
φ̂λ(x)e−π|x|
2
Hm(
√
2πx) = i−|m|e−π|λ|
2
Hsym
m (
√
2πλ), (13.8)∫
Rn
ϕλ(x)e−π|x|
2
Hm(
√
2πx) = i−|m|e−π|λ|
2
Hanti
m (
√
2πλ), (13.9)
where φ̂λ(x) is a symmetric orbit function of An−1, given by formula (6.2) in [1], ϕλ(x) is an
antisymmetric orbit function of An−1, and λ = (λ1, λ2, . . . , λn).
The polynomials Hsym
m and Hanti
m are symmetric and antisymmetric, respectively, with respect
to the Weyl group W ≡ Sn of An−1:
Hsym
m (wλ) = Hsym
m (λ), Hanti
m (wλ) = (detw)Hanti
m (λ), w ∈ Sn.
For this reason, Hsym
m (λ) are uniquely determined by their values of λ = (λ1, λ2, . . . , λn) such
that λ1 ≥ λ2 ≥ · · · ≥ λn and Hanti
m (λ) by their values of λ such that λ1 > λ2 > · · · > λn. (Note
that Hanti
m (λ) = 0 if λi = λi+1 for some i = 1, 2, . . . , n− 1.)
The polynomials Hsym
m are of the form
Hsym
m (λ) =
∑
w∈Sn
Hwm(λ), (13.10)
where the polynomials Hwm(λ) are of the form (13.5). The polynomials Hanti
m are of the form
Hanti
m (λ) =
∑
w∈Sn
(detw)Hwm(λ), (13.11)
that is,
Hanti
m (λ) = det (Hmi(λj))
n
i,j=1 .
Moreover, Hanti
m (λ) = 0 if mi = mi+1 for some i = 1, 2, . . . , n − 1. For this reason, we may
consider the polynomials Hsym
m (λ) for n-tuples m such that m1 ≥ m2 ≥ · · · ≥ mn and the
polynomials Hanti
m (λ) for n-tuples m such that m1 > m2 > · · · > mn.
Let us apply symmetric orbit function transform (8.10) of [1] to the symmetric function
(13.10). Taking into account formula (13.8) we obtain
F
(
e−π|x|
2
Hsym
m (
√
2πx)
)
:=
1
|Sn|
∫
Rn
φ̂λ(x)e−π|x|
2
Hsym
m (
√
2πx)dx
= i−|m|e−π|λ|
2
Hsym
m (
√
2πλ),
where |Sn| is an order of the permutation group Sn, that is, functions (13.10) are eigenfunctions
of the symmetric orbit function transform F. Since the functions (13.10) for mi = 0, 1, 2, . . . ,
80 A. Klimyk and J. Patera
i = 1, 2, . . . , n, m1 ≥ m2 ≥ · · · ≥ mn, form an orthogonal basis of the Hilbert space L2
sym(Rn)
of functions from L2(Rn) symmetric with respect to W , then they constitute a complete set of
eigenfunctions of this transform. Thus, this transform has only four eigenvalues i, −i, 1, −1 in
L2
sym(Rn). This means that, as in the case of the usual Fourier transform, we have
F4 = 1.
Now we apply antisymmetric orbit function transform (10.12) to the antisymmetric func-
tion (13.11). Taking into account formula (13.9) we obtain
F̃
(
e−π|x|
2
Hanti
m (
√
2πx)
)
:=
1
|Sn|
∫
Rn
ϕλ(x)e−π|x|
2
Hanti
m (
√
2πx)dx
= i−|m|e−π|λ|
2
Hanti
m (
√
2πλ), m1 > m2 > · · · > mn ≥ 0,
that is, functions (13.11) are eigenfunctions of the symmetric orbit function transform F̃. Since
the functions (13.11) for mi = 0, 1, 2, . . . ; i = 1, 2, . . . , n, m1 > m2 > · · · > mn ≥ 0, form an
orthogonal basis of the Hilbert space L2
anti(Rn) of functions from L2(Rn) antisymmetric with
respect to W , then they constitute a complete set of eigenfunctions of this transform. Thus, this
transform has only four eigenvalues i, −i, 1, −1. This means that, as in the case of the usual
Fourier transform, we have
F̃4 = 1.
13.3 Symmetric and antisymmetric sets of polynomials
In the previous subsection we constructed symmetric and antisymmetric sets of functions con-
nected with Hermite polynomials. Similarly other sets of orthogonal polynomials can be con-
structed (see [37, 38] and [39]).
Let pm(x), m = 0, 1, 2, . . . , be the set of orthogonal polynomials in one variable such that∫
R
pm(x)pm′(x)dσ(x) = δmm′ ,
where dσ(x) is some orthogonality measure, which may be finite or discrete.
We create a set of symmetric polynomials of n variables as follows:
psym
m (x) =
∑
w∈Sn/Sm
pmw(1)
(x1)pmw(2)
(x2) · · · pmw(n)
(xn), (13.12)
mi = 0, 1, 2, . . . , i = 1, 2, . . . , n,
where m = (m1,m2, . . . ,mn), m1 ≥ m2 ≥ · · · ≥ mn ≥ 0, x = (x1, x2, . . . , xn), and w(1), w(2),
. . . , w(n) is a set of numbers 1, 2, . . . , n transformed by the permutation w ∈ Sn/Sm, where Sm
is the subgroup of Sn consisting of elements leaving m invariant.
We also create the set of polynomials
panti
m (x) =
∑
w∈Sn
(detw)pmw(1)
(x1)pmw(2)
(x2) · · · pmw(n)
(xn) = det (pmi(xj))
n
i,j=1 , (13.13)
mi = 0, 1, 2, . . . , i = 1, 2, . . . , n,
where notations are the same as in (13.12) and the condition m1 > m2 > · · · > mn ≥ 0 is
satisfied.
Antisymmetric Orbit Functions 81
It is easy to check that the polynomials psym
m (x) are symmetric with respect to transformations
of Sn:
psym
m (wx) = psym
m (x), w ∈ Sn.
Similarly, the polynomials panti
m (x) are antisymmetric with respect to transformations of Sn:
panti
m (wx) = (detw)panti
m (x), w ∈ Sn.
Thus, we may consider the polynomials (13.12) and (13.13) on the closure of the fundamental
domain of the transformation group W (An−1) ≡ Sn. This closure (we denote it by D+) coincides
with the set of points x = (x1, x2, . . . , xn) for which
x1 ≥ x2 ≥ · · · ≥ xn ≥ 0.
The polynomials (13.13) vanish if for some i, i = 1, 2, . . . , n− 1, we have xi = xi+1.
The set of polynomials (13.12), as well as the set of the polynomials (13.13), is orthogonal
with respect to the product measure dσ(x) ≡ dσ(x1) dσ(x2) · · · dσ(xn). Indeed, we have∫
D+
psym
m (x)psym
m′ (x)dσ(x) =
|O(m)|
|Sn|
δmm′ =
1
|Sm|
δmm′ ,∫
D+
panti
m (x)panti
m′ (x)dσ(x) = δmm′ ,
where O(m) is the Sn-orbit of the point m.
Note that each polynomial panti
m (x) vanishes at xi = xj for any admitted i and j. This means
that panti
m (x) can be divided by xi − xj . Therefore, the functions
P anti
m (x) =
panti
m (x)∏
1≤i<j≤n
(xi − xj)
are polynomials in xi, i = 1, 2, . . . , n. These polynomials are also orthogonal and the orthogo-
nality relation is of the form∫
D+
P anti
m (x)P anti
m′ (x)Ξ(x)dσ(x) = δmm′ ,
where Ξ(x) =
∏
1≤i<j≤n
(xi − xj)
2.
Acknowledgements
The first author (AK) acknowledges CRM of University of Montreal for hospitality when this
paper was under preparation. His research was partially supported by Grant 10.01/015 of the
State Foundation of Fundamental Research of Ukraine. We are grateful for partial support for
this work to the National Research Council of Canada, MITACS, the MIND Institute of Costa
Mesa, California, and Lockheed Martin, Canada.
82 A. Klimyk and J. Patera
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1 Introduction
2 Weyl groups and affine Weyl groups
2.1 Coxeter--Dynkin diagrams and Cartan matrices
2.2 Weyl group
2.3 Roots and weights
2.4 Highest root
2.5 Affine Weyl groups
2.6 Fundamental domain
3 Weyl group signed orbits
3.1 Signed orbits
3.2 Signed orbits of A1, A1A1, A2, C2, G2
3.3 The case of An
3.4 The case of Bn
3.5 The case of Cn
3.6 The case of Dn
3.7 Signed orbits of A3
3.8 Signed orbits of B3
3.9 Signed orbits of C3
4 Antisymmetric orbit functions
4.1 Definition
4.2 Antisymmetric orbit functions of A2
4.3 Antisymmetric orbit functions of C2 and G2
4.4 Antisymmetric orbit functions of An
4.5 Antisymmetric orbit functions of Bn
4.6 Antisymmetric orbit functions of Cn
4.7 Antisymmetric orbit functions of Dn
4.8 Symmetric orbit functions of Bn, Cn and Dn
5 Properties of antisymmetric orbit functions
5.1 Anti-invariance with respect to the Weyl group
5.2 Anti-invariance with respect to the affine Weyl group
5.3 Continuity and vanishing
5.4 Realness and complex conjugation
5.5 Scaling symmetry
5.6 Duality
5.7 Orthogonality
5.8 Orthogonality to symmetric orbit functions
5.9 Antisymmetric orbit functions
5.10 Symmetric orbit functions
6 Properties of antisymmetric orbit functions of An
6.1 Decomposition of symmetric powers of representations
6.2 Properties of antisymmetric orbit functions of An
7 Decomposition of products of (anti)symmetric orbit functions
7.1 Products of symmetric and antisymmetric orbit functions
7.2 Products of symmetric and antisymmetric orbits
7.3 Products of antisymmetric orbits
7.4 Decomposition of products for rank 2
8 Decomposition of antisymmetric W-orbit functions into antisymmetric W'-orbit functions
8.1 Introduction
8.2 Decomposition of signed WAn-orbits into WAn-1-orbits
8.3 Decomposition of signed WAn-1-orbits into WAp-1WAq-1-orbits, p+q=n
8.4 Decomposition of signed WBn-orbits into WBn-1-orbits and of signed WCn-orbits into WCn-1-orbits
8.5 Decomposition of signed WCn-orbits into WAp-1WCq-orbits, p+q=n
8.6 Decomposition of signed WDn-orbits into signed WDn-1-orbits
8.7 Decomposition of signed WDn-orbits into WAp-1WDq-orbits, p+q=n, q4
9 Characters of representations and antisymmetric orbit functions
9.1 Connection of characters with orbit functions
9.2 Orthogonality of characters
9.3 Relations for antisymmetric orbit functions
10 Antisymmetric orbit function transforms
10.1 Decomposition in antisymmetric orbit functions on F
10.2 Symmetric and antisymmetric multivariate sine and cosine series
10.3 Orbit function transform on the dominant Weyl chamber
10.4 Symmetric and antisymmetric multivariate sine and cosine integral transforms
11 Multivariate generalization of the finite Fourier transform and of finite sine and cosine transforms
11.1 Finite Fourier transform
11.2 W-invariant lattices
11.3 Grids FM for A2, C2 and G2
11.4 Expanding in antisymmetric orbit functions through finite sets
11.5 Antisymmetric and symmetric multivariate discrete Fourier transforms
11.6 Discrete sine and cosine transforms
11.7 Antisymmetric multivariate discrete sine transforms
11.8 Symmetric multivariate discrete cosine transforms
11.9 Other discrete cosine transforms
11.10 Other antisymmetric multivariate discrete cosine transforms
11.11 Other symmetric multivariate discrete cosine transforms
12 Solutions of the Laplace equation on n-dimensional simplexes
12.1 The case of n-dimensional simplexes related to An, Bn, Cn and Dn
12.2 The Laplace operator in -basis
12.3 Rank two and three special cases
12.4 Antisymmetric orbit functions as eigenfunctions of other operators
13 Symmetric and antisymmetric functions
13.1 Symmetrization and antisymmetrization by (anti)symmetric orbit functions
13.2 Eigenfunctions of (anti)symmetric orbit function transform for W(An)
13.3 Symmetric and antisymmetric sets of polynomials
References
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