Representations of the Lie Superalgebra (1|2n) with Polynomial Bases
We study a particular class of infinite-dimensional representations of (1|2). These representations ₙ() are characterized by a positive integer p, and are the lowest component in the p-fold tensor product of the metaplectic representation of (1|2). We construct a new polynomial basis for ₙ() arising...
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| description | We study a particular class of infinite-dimensional representations of (1|2). These representations ₙ() are characterized by a positive integer p, and are the lowest component in the p-fold tensor product of the metaplectic representation of (1|2). We construct a new polynomial basis for ₙ() arising from the embedding (1|2) ⊃ (1|2). The basis vectors of ₙ() are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra-valued polynomials with integer coefficients in variables. Using combinatorial properties of these tableau vectors, it is deduced that they form a basis. The computation of matrix elements of a set of generators of (1|2) on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 031, 27 pages
Representations of the Lie Superalgebra osp(1|2n)
with Polynomial Bases
Asmus K. BISBO a, Hendrik DE BIE b and Joris VAN DER JEUGT a
a) Department of Applied Mathematics, Computer Science and Statistics, Ghent University,
Krijgslaan 281-S9, B-9000 Gent, Belgium
E-mail: Asmus.Bisbo@UGent.be, Joris.VanderJeugt@UGent.be
b) Department of Electronics and Information Systems, Faculty of Engineering and Architecture,
Ghent University, Krijgslaan 281-S8, B-9000 Gent, Belgium
E-mail: Hendrik.DeBie@Ugent.be
Received June 30, 2020, in final form March 10, 2021; Published online March 25, 2021
https://doi.org/10.3842/SIGMA.2021.031
Abstract. We study a particular class of infinite-dimensional representations of osp(1|2n).
These representations Ln(p) are characterized by a positive integer p, and are the lowest com-
ponent in the p-fold tensor product of the metaplectic representation of osp(1|2n). We con-
struct a new polynomial basis for Ln(p) arising from the embedding osp(1|2np) ⊃ osp(1|2n).
The basis vectors of Ln(p) are labelled by semi-standard Young tableaux, and are expressed
as Clifford algebra valued polynomials with integer coefficients in np variables. Using com-
binatorial properties of these tableau vectors it is deduced that they form indeed a basis.
The computation of matrix elements of a set of generators of osp(1|2n) on these basis vectors
requires further combinatorics, such as the action of a Young subgroup on the horizontal
strips of the tableau.
Key words: representation theory; Lie superalgebras; Young tableaux; Clifford analysis;
parabosons
2020 Mathematics Subject Classification: 17B10; 05E10; 81R05; 15A66
1 Introduction
In representation theory of Lie algebras, Lie superalgebras or their deformations, there are often
three problems to be tackled. The first is the existence or the classification of representations.
The second is obtaining character formulas for representations. And the third is the construction
of (a class of) representations. Concretely, this third step consists of finding an explicit basis for
the representation space and the explicit action of a set of algebra generators in this basis (i.e.,
find all matrix elements). Mathematicians are primarily interested in the first two problems
and often ignore the third one. This goes together with the impression that the third problem
is computationally quite hard and not necessarily leads to interesting mathematical structures.
For applications in physics however, the third step is often indispensable, as one needs to compute
physical quantities such as energy spectra coming from eigenvalues of Hamiltonians, or transition
matrix elements coming from explicit actions, e.g., [15]. A typical example of the third problem
is the construction of the Gelfand–Zetlin basis for finite-dimensional irreducible representations
of the Lie algebra gl(n) or sl(n), see, e.g., [25].
In this paper we are dealing with a class of representations of the orthosymplectic Lie super-
algebra osp(1|2n). The representations considered here are infinite-dimensional irreducible low-
est weight representations, with lowest weight coordinates
(p
2 ,
p
2 , . . . ,
p
2
)
, for p a positive inte-
ger [1, 8, 9]. As we shall see in the paper, the construction of basis vectors and generator actions
leads to many interesting mathematical and combinatorial concepts.
mailto:Asmus.Bisbo@UGent.be
mailto:Joris.VanderJeugt@UGent.be
mailto:Hendrik.DeBie@Ugent.be
https://doi.org/10.3842/SIGMA.2021.031
2 A.K. Bisbo, H. de Bie and J. Van der Jeugt
The Lie superalgebra osp(1|2n) plays an important role in mathematical physics. For n = 2,
this superalgebra describes the supersymmetric extension in D = 4 anti-de Sitter space ([7], see
also [16] and references therein). Supersymmetric higher spin extensions of the anti-de Sit-
ter algebra in four dimensions are realized in other orthosymplectic superalgebras, including
osp(1|2n) and osp(m|2n) [29, 31]. Our own motivation for studying the class of osp(1|2n)
representations mentioned in the previous paragraph also comes from physics: this class of rep-
resentations corresponds to the so-called paraboson Fock spaces. Parabosons were introduced
by Green [12] in 1953, as generalizations of bosons. Parabosons have been of interest in quan-
tum field theory [27], in generalizations of quantum statistics [2, 14] and in Wigner quantum
systems [19, 21]. Whereas creation and annihilation operators of bosons satisfy simple commu-
tation relations, those of parabosons satisfy more complicated triple relations. Moreover where
there is only one boson Fock space, there are an infinite number of paraboson Fock spaces, each
of them characterized by a positive integer p (called the order of statistics). Many years after
their introduction, it was shown that the triple relations for n pairs of paraboson operators are in
fact defining relations for the Lie superalgebra osp(1|2n) [11] and that the paraboson Fock space
of order p coincides with the unitary irreducible lowest weight representation Ln(p) of osp(1|2n)
with lowest weight
(p
2 ,
p
2 , . . . ,
p
2
)
in the natural basis of the weight space.
The construction of a convenient basis for Ln(p), with the explicit action of the paraboson
operators, turns out to be a difficult problem. In principle, one can follow Green’s approach
[12, 18] and identify Ln(p) as an irreducible component in the p-fold tensor product of the boson
Fock space of osp(1|2n) (which is Ln(1)). However, there are computational difficulties to finding
a proper basis this way.
Some years ago [22], a solution was found for the construction of a basis for Ln(p), using
in particular the embedding osp(1|2n) ⊃ gl(n). This allowed the construction of a proper
Gelfand–Zetlin basis for Ln(p), and using further group theoretical techniques the actions of the
paraboson operators in this basis could be computed [22]. This offered a solution to a long-
standing problem. A disadvantage of this solution and the Gelfand–Zetlin basis is the rather
complicated expressions for the generator matrix elements, involving square roots and elaborate
Clebsch–Gordan coefficients. For the basis constructed in this paper, all generator matrix ele-
ments are integers, calculated through combinatorial considerations. Moreover, the individual
basis elements of our construction are described by concrete polynomial expressions. This is
another difference with the Gelfand–Zetlin basis, which is essentially just a labelling of basis
vectors without a concrete functional description.
In the present paper, we construct a new polynomial basis for these representations Ln(p).
Starting from the embedding osp(1|2np) ⊃ osp(1|2n), Ln(p) can be identified with a submo-
dule of the decomposition of the osp(1|2np) Fock space Lnp(1). Equivalently, the Howe dual
pair (osp(1|2n),Pin(p)) for this Fock space can be employed. Using furthermore the character
for Ln(p), this yields a polynomial basis consisting of vectors ωA(p), labelled by semi-standard
Young tableaux A of length at most p with entries 1, 2, . . . , n. The construction of this basis is
carefully developed in Sections 2–5. Each basis vector ωA(p) is a specific Clifford algebra valued
polynomial with integer coefficients in np variables xi,α (i = 1, . . . , n; α = 1, . . . , p), thus invol-
ving p Clifford elements eα. The osp(1|2n) generators Xi and Di are realized as (Clifford algebra
valued) multiplication operators or differentiation operators with respect to the xi,α’s. These
are given in Section 2, where the definition of osp(1|2n) is recalled and Ln(p) is defined. In Sec-
tion 3 a module V n(p) is constructed, which is a natural induced osp(1|2n) module. A useful
homomorphism Ψp : V n(p) → Ln(p) is considered, yielding the conclusion that Ln(p) is a quo-
tient module of V n(p). In Section 4 we introduce the above mentioned tableau vectors ωA(p)
as candidate basis vectors for Ln(p). The same is done for tableau vectors vA(p) of the mod-
ule V n(p). In this section, the knowledge of the characters of Ln(p) and V n(p) in terms of Schur
polynomials plays an essential role. In order to show that the tableau vectors ωA(p) consti-
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 3
tute indeed a basis for Ln(p), their linear independence must be shown, and this is established
in Section 5. The proof is non-trivial and depends on a total ordering for the set of semistan-
dard Young tableaux with entries from {1, 2, . . . , n}. This allows the identification of a unique
“leading term” in ωA(p) which does not appear in ωB(p) if B < A. In Section 6 we compute
the action of the osp(1|2n) generators Xi and Di on the tableau vectors ωA(p) of Ln(p). This is
rather technical: rewriting XiωA(p) or DiωA(p) as linear combinations of tableau vectors ωB(p)
is not trivial. Fortunately, also here the identification of “leading terms” is very helpful to solve
the problem. We also give some examples, making the technical parts comprehensible. Finally,
in Section 7 we consider as an example the case n = 2 in detail.
To improve the readability of the paper, we make a list of the commonly used symbols. Here
n, p ∈ N and i ∈ {1, . . . , n}.
Hn(p) Irreducible lowest weight osp(1|2n)-module with lowest weight
(p
2 , . . . ,
p
2
)
C`p Complex Clifford algebra with p positive signature generators, eα
C[Rnp] Space of polynomials in np variables, xi,α, with complex coefficients
A Space of Clifford algebra valued polynomials, A := C[Rnp]⊗ C`p
Ln(p) Realization of Hn(p) in A generated by the osp(1|2n)-action on 1 ∈ A
Xi, Di Generators of the osp(1|2n)-action on Ln(p) and A
V n(p) Induced osp(1|2n)-module with lowest weight vector |0〉 of weight
(p
2 , . . . ,
p
2
)
B+
i , B
−
i Generators of the osp(1|2n)-action on V n(p)
Mn(p) Maximal non-trivial osp(1|2n)-submodule of V n(p)
Vn(p) Realization of Hn(p) as the quotient module Vn(p) := V n(p)/Mn(p)
Ψp The map V n(p)→ Ln(p) mapping B+
i1
· · ·B+
ik
|0〉 7→ Xi1 · · ·Xik(1)
Yn Set of semistandard Young tableaux with entries in {1, . . . , n}
Yn(p) Set of semistandard Young tableaux in Yn with at most p rows
T(λ, p) Set of column distinct Young tableaux of shape λ with entries in {1, . . . , p}
E(λ, p) Set of Young tableaux of shape λ with entries in {1, . . . , p}
λA, µA Respectively the shape and weight of a tableau A ∈ Yn
ωA(p) Basis vector of Ln(p) corresponding to A ∈ Yn(p)
ω̃A(p) Normalized vector propotional to ωA(p)
vA(p) Basis vector of V n(p) corresponding to A ∈ Yn
xA,T Monomial in C[Rnp] corresponding to A ∈ Yn(p) and T ∈ E(λ, p)
eT Element in C`p corresponding to T ∈ E(λ, p).
DA Young tableau of shape λA with k’s in all entries of the k’th row, for all k
cA(γ) Coefficient of the monomial
∏n
i=1
∏p
α=1(xi,αeα)γi,α in the expansion of ω̃A(p)
2 The polynomial paraboson Fock space Ln(p)
The Lie superalgebra osp(1|2n) is usually defined as a matrix Lie superalgebra [10, 17]. It can
also be defined as a symbolic algebra with generators and relations [11]. Adopting the latter
definition osp(1|2n) is generated by 2n odd elements B+
i and B−i , for i ∈ {1, . . . , n}, satisfying
the structural relations[{
Bξ
i , B
η
j
}
, Bε
l
]
= (ε− ξ)δi,lBη
j + (ε− η)δj,lB
ξ
i , (2.1)
4 A.K. Bisbo, H. de Bie and J. Van der Jeugt
for i, j, l ∈ {1, . . . , n} and η, ε, ξ ∈ {+,−}, to be interpreted as ±1 in the algebraic relations.
The brackets [·, ·] and {·, ·} in (2.1) denote commutators and anti-commutators respectively.
We endow osp(1|2n) with a unitary structure given by the anti-involution B±i 7→ (B±i )† := B∓i .
The Cartan subalgebra h of osp(1|2n) has a basis consisting of the n commuting elements
hi =
1
2
{
B+
i , B
−
i
}
, (2.2)
for i ∈ {1, . . . , n}. Letting εi, for i ∈ {1, . . . , n}, be the corresponding dual basis for h∗, we are
able to define the lowest weight modules we are interested in. Fixing {ε1, . . . , εn} as basis for h∗,
the notation (p1, . . . , pn) will from now on be used for the weight
∑n
i=1 piεi ∈ h∗.
Definition 2.1. Given p ∈ N, let Hn(p) be the unitary irreducible lowest weight module
of osp(1|2n) of lowest weight
(p
2 , . . . ,
p
2
)
and with lowest weight vector |0〉. The actions of
the generators B+
i and B−i , for i ∈ {1, . . . , n}, of osp(1|2n) on Hn(p) are defined by relations
B−i |0〉 = 0, and
{
B+
i , B
−
j
}
|0〉 = pδi,j |0〉, (2.3)
for all i, j ∈ {1, . . . , n}.
These modules were originally introduced in the context of parastatistics with Hn(p) as the
Fock spaces of n parabosonic particles of order p. Here they are usually referred to as paraboson
Fock spaces [12, 14]. When p = 1, the parabosonic particles revert to usual bosonic particles,
and Hn(1) becomes the usual boson Fock space. From a different point of view Hn(1) is the
Hilbert space of the quantum harmonic oscillator and is for that reason also referred to as the
oscillator representation [26].
From now on we will consider n and p to be fixed positive integers. The treatment of the rep-
resentation Hn(p) will be carried out through a certain polynomial realization, the construction
of which will take up the rest of this section.
The Clifford algebra C`p is generated by p elements eα, for α ∈ {1, . . . , p}, satisfying
{eα, eβ} = 2δα,β,
for all α, β ∈ {1, . . . , p}. Let C[Rnp] denote the space of polynomials in np variables xi,α,
for i ∈ {1, . . . , n} and α ∈ {1, . . . , p}, with complex coefficients, and define
A := C[Rnp]⊗ C`p
to be its Clifford algebra valued counterpart. We will in general suppress the tensor products
when dealing with elements of A. For each non-negative integer n× p matrix γ ∈Mn,p(N0) and
each η ∈ Np0 we define the following monomials
xγ :=
n∏
i=1
p∏
α=1
x
γi,α
i,α ∈ C[Rnp] and eη := eη11 · · · e
ηp
p ∈ C`p. (2.4)
In the context of such monomials we will refer to γ as the exponent matrix. We note that
eη = eη
′
if and only if ηα ≡ η′α mod 2, for all α ∈ {1, . . . , p}. As a vector space A has a basis
consisting of the vectors xγeη, for γ ∈ Mn,p(N0) and η ∈ Zp2. The basis is orthonormal with
respect to the following canonical inner product
〈xγeη, xγ′eη′〉 := δγ,γ′δη,η′ , (2.5)
for all γ, γ′ ∈Mn,p(N0) and η, η′ ∈ Zp2.
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 5
Denoting the 2np odd generators of osp(1|2n)nnp by B±i,α we can consider C[Rnp] as an
osp(1|2n)nnp-module with action
B+
i,αx
γ := xi,αx
γ and B−i,αx
γ := ∂i,αx
γ ,
for all i ∈ {1, . . . , n}, α ∈ {1, . . . , p} and γ ∈Mn,p(N0), where ∂i,α := ∂
∂xi,α
. The relations
[∂i,α, xj,β] = δi,jδα,β,
for i, j ∈ {1, . . . , n} and α, β ∈ {1, . . . , p}, imply the structural relations (2.1) of osp(1|2n)nnp.
In fact C[Rnp] ∼= Hnp(1) as osp(1|2n)nnp modules. Similarly we can consider A as an osp(1|2n)-
module by defining operators
Xi :=
p∑
α=1
xi,αeα and Di :=
p∑
α=1
∂i,αeα, (2.6)
and letting
B+
i (xγeη) := Xi(x
γeη) =
p∑
α=1
xi,αx
γeαe
η, (2.7)
B−i (xγeη) := Di(x
γeη) =
p∑
α=1
∂i,αx
γeαe
η, (2.8)
for all i ∈ {1, . . . , n}, γ ∈ Mn,p(N0) and η ∈ Np0. It is easily verified that A is an osp(1|2n)-
module by checking that the actions (2.8) satisfy the relations (2.1). It can furthermore easily
be verified that the inner product (2.5) on A respects the unitary structure of osp(1|2n), thus
making A a unitary osp(1|2n)-module.
The subsequent result, Proposition 2.3, tells us that A has an irreducible submodule isomor-
phic to Hn(p). The shape of the operators in (2.6) is that of the Green’s ansatz [12] with the
internal components here being realized using Clifford algebra elements. This type of realization
of the Green’s ansatz dates back to [13, 14].
The module A admits the Howe dual pair [5] (osp(1|2n), G), where G = Pin(p) when p is even
and G = Spin(p) when p is odd. This gives a multiplicity free decomposition of A into a direct
sum of irreducible modules of osp(1|2n)×G of the form Hosp(µ)⊗HG(ν) [3, 4, 28]. Here Hosp(µ)
denotes an irreducible lowest weight module of osp(1|2n) of lowest weight µ, and HG(ν) denotes
an irreducible highest weight module of G of highest weight ν.
We are interested in exactly one of the irreducible modules in this decomposition, namely
the one with the constant polynomial 1 ⊗ I ∈ A as lowest weight vector. Here I ∈ C`p is the
identity element in the Clifford algebra. We shall from now on use the notation 1 := 1⊗ I ∈ A.
Definition 2.2. Let Ln(p) be the submodule of A generated by the action of osp(1|2n) on 1 ∈ A.
Proposition 2.3. The module Ln(p) is equivalent to the irreducible lowest weight osp(1|2n)-
module Hn(p).
Proof. In the module A the basis for the Cartan subalgebra of osp(1|2n), as described in (2.2),
is represented by operators
hi 7→
1
2
{Xi, Di} =
p
2
+
p∑
α=1
xi,α∂i,α,
6 A.K. Bisbo, H. de Bie and J. Van der Jeugt
for all i ∈ {1, . . . , p}. Since
Di(1) = 0 and
1
2
{Xi, Dj}(1) =
p
2
δi,j(1),
for all i, j ∈ {1, . . . , n}, it follows that 1 is a lowest weight vector of weight
(p
2 , . . . ,
p
2
)
. As men-
tioned above, the module A decomposes into a direct sum of irreducible lowest weight modules
of osp(1|2n) as a result of admitting the Howe dual pair (osp(1|2n), G). It follows that Ln(p)
must be one of the components in the decomposition and hence be irreducible. It can thus be
concluded that Ln(p) is equivalent to Hn(p). �
In the following sections we will construct a basis for Ln(p) together with formulas for the
action of osp(1|2n) on the basis.
3 The induced module V n(p)
In this section we introduce the induced module V n(p) which was used in [22] to study the
paraboson Fock space. We relate this module to Ln(p) via a surjective module homomorphism Ψp
and prove that the kernel of this map can be identified with the maximal non-trivial submodule
of V n(p). The relationship between V n(p) and Ln(p) described by Ψp will be used throughout
the following chapters and serves as a valuable tool in the proof of Theorem 5.7.
Following [22] we consider the parabolic subalgebra of osp(1|2n) given by
P = span
{{
B+
i , B
−
j
}
, B−i , {B
−
i , B
−
j } : i, j ∈ {1, . . . , n}
}
.
Let |0〉 describe the lowest weight vector of the module Hn(p), then the action (2.3) of P on |0〉
generates a one dimensional P-module denoted C|0〉. As in [22] we let V n(p) denote the induced
module of osp(1|2n) relative to P, see [10],
V n(p) := Ind
osp(1|2n)
P C|0〉.
We let Vn(p) denote the irreducible module obtained by taking the quotient of V n(p) by its
maximal non-trivial submodule Mn(p),
Vn(p) := V n(p)/Mn(p).
By construction Vn(p) is then isomorphic to Hn(p) as an osp(1|2n)-module.
Let osp(1|2n)+ denote the subalgebra of osp(1|2n) generated by B+
i , for i ∈ {1, . . . , n},
osp(1|2n)+ := span
{
B+
i ,
{
B+
i , B
+
j
}
: i, j ∈ {1, . . . , n}
}
.
As a vector space osp(1|2n) = osp(1|2n)+⊕P. The definition of the induced module V n(p) then
implies the following result that will be used in the proof of Theorem 5.7.
Lemma 3.1. The map
Φp : U(osp(1|2n)+)→ V n(p), B 7→ B|0〉,
is an isomorphism of vector spaces.
We relate the modules V n(p) and Ln(p) by the module homomorphism
Ψp : V n(p)→ Ln(p),
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 7
for which Ψp(|0〉) := 1. This means that Ψp(B
+
i v) = XiΨp(v), and Ψp(B
−
i v) = DiΨp(v), for all
i ∈ {1, . . . , n} and v ∈ V n(p). To continue, the following notation is needed. For each k ∈ N,
consider the index set
I(k) :=
{
(i1, . . . , ik) ∈ {1, . . . , n}k : i1 < · · · < ik
}
.
To each I = (i1, . . . , ik) ∈ I(k) we assign the following antisymmetric sums of operators acting
on Ln(p) and V n(p) respectively.
XI :=
∑
σ∈Sk
sgn(σ)Xiσ(1) · · ·Xiσ(k) (3.1)
and
B+
I :=
∑
σ∈Sk
sgn(σ)B+
iσ(1)
· · ·B+
iσ(k)
, (3.2)
where Sk denotes the k’th symmetric group. Whereas B+
I 6= 0, for all I ∈ I(k) and k ∈ N,
it holds that XI = 0 if k > p and XI 6= 0 if k ≤ p. This is verified by a short calculation showing
that
XI = k!
∑
α1,...,αk∈{1,...,p},
αi 6=αj , when i 6=j
xi1,α1 · · ·xik,αkeα1 · · · eαk , (3.3)
where the sum is over all sets of k mutually distinct integers in {1, . . . , p}.
Lemma 3.2. The map Ψp is a surjective osp(1|2n)-module homomorphism with kernel
ker Ψp = Mn(p)
generated by the action of osp(1|2n) on the vectors B+
I |0〉 with I ∈ I(p+ 1).
Proof. The surjectivity of the osp(1|2n)-module homomorphism Ψp follows essentially from
the way it is constructed. The discussion preceding (3.3) implies that B+
I |0〉 ∈ ker Ψp, for all
I ∈ I(p + 1), and that these are non-zero vectors. The maximality of Mn(p) implies that
ker Ψp ⊂Mn(p), and thus that B+
I |0〉 ∈Mn(p), for all I ∈ I(p+ 1).
From the construction in [22] of a Gelfand–Zetlin basis for V n(p) it follows that Mn(p) is
generated by the action of osp(1|2n) on the Gelfand–Zetlin basis vectors with top row
(1, 1, . . . , 1︸ ︷︷ ︸
p+1
, 0, 0, . . . , 0),
in their Gelfand–Zetlin labelling. These vectors can be naturally indexed by the set I(p+1) such
that the vector with index I ∈ I(p+ 1) has weight εI := p
2
∑n
i=1 εi+
∑p+1
k=1 εik . The weight space
of Mn(p) corresponding to the weight εI is 1-dimensional and B+
I |0〉 is a vector of weight εI . This
means that Mn(p) is generated by the action of osp(1|2n) on the vectors B+
I |0〉, for I ∈ I(p+1),
and thus that ker Ψp = Mn(p). �
Recalling that Hn(p) ∼= Vn(p) = V n(p)/Mn(p) this result implies Proposition 2.3. Further-
more it implies that V n(p) ∼= Ln(p) when p ≥ n, since in that case I(p + 1) = ∅ and thus
Mn(p) = {0}. This observation is in fact closely related to a more general result [6], [20, The-
orem 1.1]. In the context of the present paper the result can be stated as follows. Let W be
any irreducible factor in the decomposition of A. Denote the lowest weight of W by µ ∈ h∗
and let V(µ) denote the corresponding induced module, constructed in much the same way as
we constructed V n(p). Then W ∼= V(µ) whenever p ≥ 2n. As mentioned above this condition
reduces to p ≥ n for the modules we study here, namely the paraboson Fock spaces. This is a
consequence of the fact that they have a unique vacuum, contrary to a general irreducible factor
of A.
8 A.K. Bisbo, H. de Bie and J. Van der Jeugt
4 Tableau vectors in Ln(p) and V n(p)
The main result of this paper is the explicit construction of bases for the modules Ln(p)
and V n(p) which are related by the map Ψp. In this section we will construct the sets of
vectors that will be our candidates for the desired bases, see Definition 4.2. Proving that they
are indeed bases for the modules is postponed to the next section. As touched upon in Section 3,
bases for the modules Vn(p) and V n(p), which are related by the quotient map
V n(p)→ V n(p)/Mn(p) = Vn(p),
are found and studied in the paper [22]. These basis vectors are identified only with their
labellings by Gelfand–Zetlin patterns relative to the gl(n) subalgebra of osp(1|2n). The new
basis constructed in this section, on the contrary, will consist of explicit polynomials in Ln(p).
Partitions, Young tableaux and symmetric functions will be used throughout this paper.
In the interest of consistency, we will be using the notation established in Macdonald [24].
Let the set of all partitions be denoted by P, and let `(λ) be the length of the partition λ ∈ P.
Let Yn denote the set of semistandard (s.s.) Young tableaux with weights in Nn0 , that is with
numbers in the set {1, . . . , n} as entries. Let λA ∈ P and µA ∈ Nn0 denote the shape and weight
of A ∈ Yn respectively and let
Yn(p) = {A ∈ Yn : `(λA) ≤ p}
be the set of s.s. Young tableaux in Yn with at most p rows. S.s. Young tableaux are enumerated
by Kostka numbers
Kλ,µ := #{s.s. Young tableaux of shape λ and weight µ}. (4.1)
The first use we shall make of this terminology is to describe the weight space dimensions
of Ln(p) and V n(p) as sums of Kostka numbers. This description will serve as the main inspira-
tion for defining basis vectors for the modules. The action of B±i on any weight vector of Ln(p)
or V n(p) changes its weight by ±εi, for any i ∈ {1, . . . , n}. Therefore, any weight corresponding
to a non-zero weight space of either module will be of the form
µ+
p
2
:= µ+
(
p
2
, . . . ,
p
2
)
∈ h∗,
for some µ ∈ Nn0 . We will use the notations V n(p)µ+ p
2
and Ln(p)µ+ p
2
respectively for the weight
spaces of V n(p) and Ln(p) corresponding to the weight µ+ p
2 , for any µ ∈ Nn0 .
Lemma 4.1. For any µ ∈ Nn0 ,
dimV n(p)µ+ p
2
=
∑
λ∈P
Kλ,µ, and dimLn(p)µ+ p
2
=
∑
λ∈P,
`(λ)≤p
Kλ,µ. (4.2)
Proof. The character formulas of V n(p) and Vn(p) were obtained in the papers [23], [22, Theo-
rem 7]. Recalling that Vn(p) ∼= Ln(p) these character formulas are
charV n(p) = (t1 · · · tn)
p
2
∑
λ∈P
sλ(t1, . . . , tn),
and
charLn(p) = (t1 · · · tn)
p
2
∑
λ∈P,
`(λ)≤p
sλ(t1, . . . , tn),
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 9
where sλ denotes the Schur function indexed by λ ∈ P, and ti denotes the formal exponential eεi .
The lemma then follows by recalling two facts. First, in the character formulas the coefficients
of monomials t
µ1+
p
2
1 · · · tµn+
p
2
n , for µ ∈ Nn0 , describe the dimensions of the weight spaces of the
relevant modules corresponding to the weight µ + p
2 . Second, by [30], the monomial expansion
of the Schur function indexed by λ ∈ P is
sλ(t1, . . . , tn) =
∑
µ∈Nn0
Kλ,µt
µ1
1 · · · t
µn
n . �
Recalling how the Kostka numbers enumerate s.s. Young tableaux we may interpret Lem-
ma 4.1 as saying that the modules V n(p) and Ln(p) admit bases indexed by the tableaux in Yn
and Yn(p) respectively in such a way that the weights of the s.s. Young tableaux are related to
the weights of their corresponding basis vectors by the addition of
(p
2 , . . . ,
p
2
)
.
The existence of such bases is not significant in itself unless we can find relatively simple
descriptions of the vectors in terms of the corresponding tableaux. For the construction of the
basis vectors for Ln(p) and V n(p) the idea is to interpret each column of a given s.s. Young
tableau A ∈ Yn as an operator of the form (3.1) or (3.2) and then let each such operator act
on the lowest weight vector with the leftmost column acting first.
To formalize this we need some notation. Given a partition λ ∈ P, we denote the conjugate
partition by λ′. Let al denote the l’th column, counted from left to right, of A ∈ Yn, for all
l ∈ {1, . . . , `(λ′A)}. Since the entries of the columns of A are strictly increasing downwards we
can naturally consider al as an element of I((λ′A)l), for all l ∈ {1, . . . , `(λ′A)}, noting here that
the amount of columns in A is indeed `(λ′A).
Definition 4.2. Given A ∈ Yn we let m = `(λ′A) and define
vA(p) := B+
am · · ·B
+
a1 |0〉 ∈ V n(p), and ωA(p) := Xam · · ·Xa1(1) ∈ Ln(p).
In Section 5 we will prove that these vectors form bases for V n(p) and Ln(p) respectively.
There it will be useful to know that they satisfy the following properties.
Lemma 4.3. Let A ∈ Yn, then vA(p) and ωA(p) both have weight µA + p
2 . Furthermore
Ψp(vA(p)) =
{
ωA(p), if A ∈ Yn(p),
0, if A /∈ Yn(p).
In order to illustrate the above-mentioned concepts, consider an example. Let n = 5, λA =
(4, 3, 1), µA = (2, 2, 3, 1, 0) and
A =
1 1 2 3
2 3 3
4
The columns of A are then a1 = (1, 2, 4), a2 = (1, 3), a3 = (2, 3) and a4 = (3). In that case
vA(p) = B+
3 B
+
(2,3)B
+
(1,3)B
+
(1,2,4)|0〉 and ωA(p) = X3X(2,3)X(1,3)X(1,2,4)(1),
where B+
al
and Xal , for l ∈ {1, . . . , 4}, are defined in (3.2) and (3.1) respectively. In the case
p ≥ 3, vA(p) 6= 0, ωA(p) 6= 0. On the other hand, if p < 3, then vA(p) 6= 0 and ωA(p) = 0.
10 A.K. Bisbo, H. de Bie and J. Van der Jeugt
5 Bases for Ln(p) and V n(p)
The goal of this section is to prove that the sets {vA(p) : A ∈ Yn} and {ωA(p) : A ∈ Yn(p)},
defined in Definition 4.2, form bases for V n(p) and Ln(p). This is the content of Theorem 5.7.
The main difficulty lies in proving mutual linear independence of the vectors in {ωA(p) : A ∈
Yn(p)}, the rest is achieved by applying results from the previous sections. To prove this linear
independence we construct a total ordering of Yn(p) with the following property
ωA(p) /∈ span{ωB(p) : B ∈ Yn(p), B < A}.
Given that the weight spaces of Ln(p) are finite dimensional by Lemma 4.1, the existence of such
a total ordering implies linear independence.
Definition 5.1. Given A ∈ Yn and k ∈ {1, . . . , n} the k’th subtableau of A is defined to be the
tableau Ak ∈ Yn obtained by truncating A to only the entries containing the numbers 1, . . . , k.
For example, we have
A =
1 1 2 4
2 3 4
=⇒ A4 = A, A3 =
1 1 2
2 3
, A2 =
1 1 2
2
and A1 = 1 1 .
To define the total ordering of Yn we endow both Nn0 and P with the graded lexicographic
ordering, both denoted by <. The definition of these orderings are given in the Appendix A.
Definition 5.2. Given A,B ∈ Yn we write A < B if µA < µB in Nn0 , or if µA = µB and there
exists k ∈ {1, . . . , n} such that, for all l < k,
λAl = λBl and λAk < λBk in P.
The relation < on Yn defined above is a total order. This follows from the fact that the
graded lexicographic order gives a total ordering of both Nn0 and P. The total ordering of Yn
defined above is inherited by Yn(p).
To get a feel for this ordering of Yn consider the following example. Let n = 5, and consider
the ascending chain of all 13 s.s. Young tableaux of weight µ = (2, 1, 1, 1, 0):
1 1
2
3
4
<
1 1
2 4
3
<
1 1 4
2
3
<
1 1
2 3
4
<
1 1 4
2 3
<
1 1 3
2
4
<
1 1 3
2 4
<
1 1 3 4
2
<
1 1 2
3
4
<
1 1 2
3 4
<
1 1 2 4
3
<
1 1 2 3
4
< 1 1 2 3 4 .
To see the connection between this ordering and the prospective basis for Ln(p) we need to
consider the expansion of the vectors ωA(p) into monomials. To do so, a new class of tableaux
is needed. Let
T(λ, p) :=
{
Fillings of the Young diagram of shape λ ∈ P with
numbers 1, . . . , p occuring at most once in each column
}
.
We will refer to the tableaux in T(λ, p) as column distinct (c.d.) Young tableaux. Some examples
of these are:
2 1 2 1
3 3 3 3
1 4
,
3 1 2 1
2 4 3 3
1 3
,
3 1 3 3
2 4 2 1
1 3
∈ T((4, 4, 2), 4).
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 11
From now on we shall adopt the slightly abusive notation of writing λ both for a partition
in P and for the set of coordinates in the corresponding Young diagram. That is,
λ =
{
(k, l) : l ∈ {1, . . . , `(λ′)} and k ∈ {1, . . . , λ′l}
}
,
for all λ ∈ P. With the notion of c.d. Young tableaux the following monomials in C[Rnp] and C`p
can be defined. Given a tableau C ∈ T(λ, p) and A ∈ Yn(p) of shape λ = λA ∈ P. We denote
the entries of A and C by ak,l and ck,l respectively, for all (k, l) ∈ λ. Letting m = `(λ′) be the
number of columns in A and C we define monomials
eC := (ec1,m · · · ecλ′m,m) · · · (ec1,1 · · · ecλ′1,1) ∈ C`p (5.1)
and
xA,C :=
∏
(k,l)∈λ
xak,l,ck,l ∈ C[Rnp]. (5.2)
For example, consider the case n = 4, p = 4, and
A =
1 2
2 3
3 4
and C =
2 3
3 1
1 4
.
The monomials defined in (5.1) and (5.2) will then take the form
eC = e3e1e4e2e3e1 = e2e4
and
xA,C = x1,2x
2
2,3x
2
3,1x4,4.
To each partition λ ∈ P we associate the following factorial:
λ! := λ′1! · · ·λ′λ1 ! ∈ N.
Recalling how ωA(p), eC and xA,C where defined in Definition 4.2, (5.1) and (5.2), respectively,
a short calculation using (3.3) gives the following monomial expansion of ωA(p).
Lemma 5.3. For all A ∈ Yn(p), we have
ωA(p) = λA!
∑
C∈T(λA,p)
xA,CeC .
To illustrate Definition 4.2 and Lemma 5.3 we consider as an example n = 2, p = 2 and
A =
1 1 2
2
Definition 4.2 gives
ωA(p) = X2X1X1,2 = 2!(x2,1e1 + x2,2e2)(x1,1e1 + x1,2e2)(x1,1x2,2e1e2 + x1,2x2,1e2e1).
On the other hand Lemma 5.3 gives
ωA(p) = 2x( 1 1 2
2
, 2 1 2
1
)e 2 1 2
1
+ 2x( 1 1 2
2
, 1 2 1
2
)e 1 2 1
2
+ 2x( 1 1 2
2
, 2 1 1
1
)e 2 1 1
1
+ 2x( 1 1 2
2
, 2 2 2
1
)e 2 2 2
1
+ 2x( 1 1 2
2
, 1 1 1
2
)e 1 1 1
2
+ 2x( 1 1 2
2
, 1 2 2
2
)e 1 2 2
2
+ 2x( 1 1 2
2
, 1 1 2
2
)e 1 1 2
2
+ 2x( 1 1 2
2
, 2 2 1
1
)e 2 2 1
1
.
12 A.K. Bisbo, H. de Bie and J. Van der Jeugt
Both formulas give the expansion
ωA(p) = −4x1,1x1,2x2,1x2,2 − 2x1,1x1,2x
2
2,1e1e2 − 2x21,2x2,1x2,2e1e2 + 2x21,1x2,1x2,2e1e2
+ 2x1,1x1,2x
2
2,2e1e2 + 2x21,1x
2
2,2 + 2x21,2x
2
2,1.
When looking at an expansion such as the one in Lemma 5.3, it is natural to ask the following
questions. Given A ∈ Yn(p) and C ∈ T(λA, p), does there exist C ′ ∈ T(λA, p) with C 6= C ′ such
that
xA,CeC = ±xA,C′eC′ .
If so, which C ′ ∈ T(λA, p) has this property. As is demonstrated by the following example, (5.3),
the answer to the first question is in some instances yes, though we will shortly describe a class
of c.d. Young tableaux for which the answer is always no, see (5.7),
x( 1 1
2 2
, 1 2
2 1
)e 1 2
2 1
= x1,2x2,1x1,1x2,2 = x( 1 1
2 2
, 2 1
1 2
)e 2 1
1 2
. (5.3)
The second question will be dealt with in more detail in Section 6. However we will make some
preliminary considerations here. For any exponent matrix γ ∈Mn,p(N0) we denote the row and
column sums as follows
µγ :=
(
p∑
α=1
γ1,α, . . . ,
p∑
α=1
γn,α
)
∈ Nn0 and ηγ :=
(
n∑
i=1
γi,1, . . . ,
n∑
i=1
γi,p
)
∈ Np0.
The definitions of eC and xA,C in (5.1) and (5.2) then imply that for each pair of tableaux
A ∈ Yn(p) and C ∈ T(λA, p) there exists a positive integer N(C) ∈ N and a unique exponent
matrix γA,C ∈Mn,p(N0) with µγA,C = µA such that
xA,CeC = (−1)N(C)xγA,CeηγA,C , (5.4)
recalling that xγ and eηγA,C were defined in (2.4). A combinatorial formula for determining
(−1)N(C) is given in (6.10). We note that
(γA,C)i,α = #{(k, l) ∈ λA : ak,l = i, ck,l = α} (5.5)
for all i ∈ {1, . . . , n} and α ∈ {1, . . . , p}. Using (2.5) we denote the normalized coefficient
of xγeηγ in ωA(p) by
cA(γ) :=
1
λA!
〈
xγeηγ , ωA(p)
〉
,
for all γ ∈ Mn,p(N0), η ∈ Np0 and A ∈ Yn(p). The expansion of ωA(p) presented in Lemma 5.3
implies the following result.
Lemma 5.4. Let A ∈ Yn(p), then
ωA(p) = λA!
∑
γ∈Mn,p(N0)
µγ=µA
cA(γ)xγeηγ ,
where
cA(γ) =
∑
C∈T(λA,p)
γA,C=γ
(−1)N(C). (5.6)
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 13
Since the coefficients cA(γ) are integer, the expansion of ωA(p) into monomials xγeηγ has
integer coefficients. As we shall see in Section 6, this is part of the reason that the generator
matrix elements turn out to be integers.
While it is true, as illustrated by (5.3), that some terms appear multiple times, possibly with
different signs, in the expansion from Lemma 5.3 other terms are more special. Specifically,
it will be proven in Proposition 5.5 that the terms corresponding to c.d. Young tableaux of
the following type are linearly independent of the rest of the terms in the expansion. Given
A ∈ Yn(p), let
DA ∈ T(λA, p) (5.7)
be the c.d. Young tableau with all 1’s in the 1st row, 2’s in the 2nd row and k’s in the kth row.
For these tableaux (5.5) implies that γA,DA is lower triangular and that
(γA,DA)i,α = #{i’s in the α’th row of A}. (5.8)
As an example of these tableaux,
if A =
1 1 2 3
2 2 3 4
3 4
, then DA =
1 1 1 1
2 2 2 2
3 3
.
Proposition 5.5. Let A,B ∈ Yn(p) with A < B and λ = λA, then
(a) cA(γA,DA) = (−1)
∑λ1
j=1
(j−1)λ′j(λ
′
j−1)
2 6= 0,
(b) cA(γB,DB ) = 0.
In other words, the expansion of ωA(p) described in Lemma 5.3 contains a unique leading term
xA,DAeDA which is linearly independent of all the other terms in the expansion. Furthermore,
if A < B, the leading term xB,DBeDB of ωB(p) does not appear in the expansion of ωA(p).
Proof. We begin with the proof of statement (a). A short calculation based on (5.4) shows that
(−1)N(DA) = (−1)
∑λ1
j=1
(j−1)λ′j(λ
′
j−1)
2 . Recalling (5.6) it is then enough to prove that γA,DA = γA,C
iff DA = C, for all C ∈ T(λA, p). One implication is trivial. To prove the other we suppose for
contradiction that there exists C ∈ T(λA, p) such that γA,DA = γA,C and DA 6= C. We shall use
the notation γ = γA,DA and ζ = γA,C for the remainder of the proof of (a).
The assumption that γ = ζ together with the formula (5.5) and the fact that γ is lower
triangular, implies that
ak,l ≥ ck,l, (5.9)
for all (k, l) ∈ λA.
Recall that the (k, l)′th entry of DA is k, for all (k, l) ∈ λA. The second assumption, namely
that DA 6= C, implies that there exists (k0, l0) ∈ λA such that ck0,l0 6= k0. Let s = ak0,l0 , then
among the possible options we may chose (k0, l0) such that s is as small as possible and k0 is
as large as possible. In other words (k0, l0) is chosen such that ck,l = k for all (k, l) ∈ λA with
ak,l < s, or with ak,l = s and k > k0. Since the entries in the l0’th column of C are distinct,
our choice of (k0, l0) together with (5.9) imply that ck0,l0 = t, for some t ∈ {k0 + 1, . . . , s}.
Furthermore, since t > k0, our choice of (k0, l0) also implies that ct,l = t for all l ∈ {1, . . . , (λA)t}
with at,l = s. By (5.8), we have thus found 1 + γs,t distinct coordinates (k, l) ∈ λA such that
ak,l = s and ck,l = t. By (5.5) this means that ζs,t ≥ 1 + γs,t and thus that γ 6= ζ. This
contradicts our initial assumption, finally proving that γA,DA = γA,C iff DA = C.
14 A.K. Bisbo, H. de Bie and J. Van der Jeugt
To prove statement (b), let A,B ∈ Yn(p) with A < B. By (5.6) it is enough to prove
that γB,DB 6= γA,C , for all C ∈ T(λA, p). This statement is trivial if µB = µA. We assume
therefore that A and B have the same weight and consider an arbitrary C ∈ T(λA, p). For the
remainder of this proof we shall use the notation γ = γB,DB and ζ = γA,C . By Definition 5.2
the assumption A < B implies that there exists s, t ∈ {1, . . . , n} such that
λAi = λBi , (λAs)α = (λBs)α and (λAs)t < (λBs)t,
for all i < s and α < t. Using this together with (5.8) and the fact that in each column of C the
entries are mutually distinct, we can make the estimate∑
i≤s,α≤t
ζiα ≤
∑
α≤t
(λAs)α <
∑
α≤t
(λBs)α =
∑
i≤s,α≤t
γiα.
This clearly implies γ 6= ζ, which proves (b). �
One should note that DA is not the only c.d. Young tableau in T(λA, p) with the properties
described in Proposition 5.5. These properties are satisfied by any tableau in T(λA, p) for which
all entries of any single row are equal.
Corollary 5.6. For any B ∈ Yn(p), ωB(p) 6= 0 and ωB(p) /∈ spanC{ωA(p) : A < B}.
We can now finally prove that the vectors defined in Definition 4.2, with properties described
in Lemma 4.3, form bases for the modules V n(p) and Ln(p).
Theorem 5.7. The modules V n(p) and Ln(p) have bases
{vA(p) : A ∈ Yn} and {ωA(p) : A ∈ Yn(p)}.
Proof. We begin with {ωA(p) : A ∈ Yn(p)}. It is enough to prove that we have bases for each
of the weight spaces of Ln(p). That is, it is enough to prove that, for any µ ∈ Nn0 , the set
{ωA(p) : A ∈ Yn(p), µA = µ}
is a basis for Ln(p)µ+ p
2
. By Lemma 4.3 they have the right weight. Since the set is finite, linear
independence follows from Corollary 5.6. Lemma 4.1 and (4.1) tells us that the cardinality of
the set {ωA(p) : A ∈ Yn(p), µA = µ} agrees with the dimension of the weight space, proving
that we indeed have a basis.
To prove that {vA(p) : A ∈ Yn} is a basis for V n(p) we consider q ∈ N, with q ≥ n. The com-
ments following Lemma 3.2 tell us that Ψq : V n(q) → Ln(q) is an isomorphism of osp(1|2n)-
modules. This together with Lemma 3.1 implies that the composition of Ψq and Φq,
Ψq ◦ Φq : U
(
osp(1|2n)+
)
→ Ln(q),
is an isomorphism of vector spaces. Following the notation used in Definition 4.2 we define
elements
B+
A := B+
am · · ·B
+
a1 ∈ U
(
osp(1|2n)+
)
,
for each A ∈ Yn, where m = `(λ′A). Noting that Ψq ◦Φq
(
B+
A
)
= ωA(q), for all A ∈ Yn, it follows
that the set {B+
A : A ∈ Yn} is a basis for U
(
osp(1|2n)+
)
and thus by Lemma 3.1 that{
vA(p) = B+
A |0〉 = Φp
(
B+
A
)
: A ∈ Yn
}
,
is a basis for V n(p) regardless of the value of p. �
The proof of this Theorem yields the following corollary.
Corollary 5.8. The set{
B+
A := B+
am · · ·B
+
a1 : A ∈ Yn
}
,
where m = `(λ′A), for each A, forms a basis for U
(
osp(1|2n)+
)
.
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 15
6 Action of osp(1|2n) on tableau vectors ωA(p)
In this section we will obtain formulas for the action of the osp(1|2n) generators Xi and Di on
the basis for Ln(p) constructed in Definition 4.2. More precisely, considering the normalization
ω̃A(p) :=
1
λA!
ωA(p),
for all A ∈ Yn(p), we want to obtain coefficients c̄B(i, p, A) and ĉB(i, p, A), for all i ∈ {1, . . . , n}
and A ∈ Yn(p), such that
Xiω̃A(p) =
∑
B∈Yn(p)
c̄B(i, p, A)ω̃B(p), (6.1)
Diω̃A(p) =
∑
B∈Yn(p)
ĉB(i, p, A)ω̃B(p). (6.2)
Regarding the actions of B+
i and B−i on the basis vectors ṽA(p) = 1
λA!
vA(p) of V n(p), the
corresponding expansion coefficients can be calculated from the coefficients in (6.2). The details
of this will be discussed at the end of this section after we obtain formulas for the calculation
of the coefficients c̄B(i, p, A) and ĉB(i, p, A), see Proposition 6.6.
Before producing general formulas for obtaining the expansions in (6.2), we take a look at
a few cases where the actions of Xi and Di on ω̃A(p) are particularly simple. First, if (µA)i = 0
then Diω̃A(p) = 0. Second, if i is greater than or equal to the topmost entry of the rightmost
column of A, then Xiω̃A(p) = ω̃B(p), where B is the tableau obtained by adding a single column
containing only the box i to the right side of A. For example, if A =
2 3
4
, then
Xiω̃ 2 3
4
(p) = ω̃ 2 3 i
4
(p),
for all i ≥ 3.
The vectors Xiω̃A(p) and Diω̃A(p) are weight vectors, for all i ∈ {1, . . . , n} and A ∈ Yn(p),
specifically
Xiω̃A(p) ∈ Ln(p)µA+εi+ p
2
and Diω̃A(p) ∈ Ln(p)µA−εi+ p
2
.
This fact leads to the following observation
Remark 6.1. For all i ∈ {1, . . . , n} and A,B ∈ Yn(p),
c̄B(i, p, A) = 0 if µB 6= µA + εi
and
ĉB(i, p, A) = 0 if µB 6= µA − εi.
Given a weight µ ∈ Nn0 we let dµ be the dimension of the weight space Ln(p)µ+ p
2
, see (4.2),
dµ := dimLn(p)µ+ p
2
.
Remark 6.1 then tells us that Xiω̃A(p) and Diω̃A(p) are linear combinations of at most dµA+εi
and dµA−εi tableau vectors respectively.
The remaining coefficients in (6.2) can be obtained by solving the system of linear equations
that comes from comparing the monomial expansions, Lemma 5.4, of Xiω̃A and Diω̃A with those
of ω̃B(p) for µB = µA + εi and µB = µA − εi respectively. This process is very inefficient as it
16 A.K. Bisbo, H. de Bie and J. Van der Jeugt
entails calculating all the coefficients in the monomial expansion of the involved vectors. Our
approach will be to reduce the number of coefficients we need to determine to only the necessary
ones and subsequently to find formulas for calculating them.
Given A ∈ Yn(p) and C ∈ T(λA, p) the coefficient of xA,CeC in the monomial expansion of
a given vector v ∈ Ln(p) is equal to
〈xA,CeC , v〉.
Given µ ∈ Nn0 we denote the dµ s.s. Young tableaux A1, . . . , Adµ of weight µ in such a way that
A1 < · · · < Adµ .
Define the dµ × dµ matrix Uµ and the vectors fµ(v) ∈ Cdµ , for all v ∈ Ln(p)µ+ p
2
, as follows
(Uµ)k,l := 〈xAk,DAk eDAk , ω̃Al(p)〉
and
(fµ)k(v) := 〈xAk,DAk eDAk , v〉,
for all k, l ∈ {1, . . . , dµ}. The (k, l)’th entry of the matrix Uµ is thus the coefficient of the leading
term xAk,DAk eDAk of ω̃Ak(p) as it appears in ω̃Al(p).
Proposition 6.2. For any µ ∈ Nn0 , the matrix Uµ ∈ Mdµ,dµ(Z) is integer and upper unitrian-
gular. Furthermore, for any v ∈ Ln(p)µ+ p
2
, we have
v =
dµ∑
k=1
(
U−1µ · fµ(v)
)
k
ω̃Ak(p)
=
dµ∑
k=1
dµ−k∑
t=0
∑
k=l0<···<lt≤dµ
(−1)t
t∏
s=1
(Uµ)ls−1,ls(fµ)ltω̃Ak(p).
Since Xiω̃A(p) and Diω̃A(p) are weight vectors this result provides formulas for calcula-
ting the coefficients c̄B(i, p, A) and ĉB(i, p, A), only dependent on UµA+ε1 and fµA+εi(Xiω̃A(p)),
and UµA−ε1 and fµA−εi(Diω̃A(p)), respectively.
In Lemma 6.3, we will see that the entries of the matrices UµA±ε1 and vectors fµA+εi(Xiω̃A(p))
and fµA−εi(Diω̃A(p)), are algebraic expressions in coefficients cB(γ). Recalling Lemma 5.4,
this means that the coefficients c̄B(i, p, A) and ĉB(i, p, A), also known as the generator matrix
elements, are integers.
Proof. The expansion in Lemma 5.3 implies that the matrix Uµ is integer, and Corollary 5.6
implies that it is upper unitriangular. Together Corollary 5.6 and Theorem 5.7 gives
〈xAk,DAk eDAk , v〉 =
dµ∑
l=1
〈xAk,DAk eDAk , ω̃Al(p)〉〈ω̃Al(p), v〉
= 〈ω̃Ak(p), v〉+
dµ∑
l=k+1
〈xAk,DAk eDAk , ω̃Al(p)〉〈ω̃Al(p), v〉.
By moving the sum to the other side of the equation and substituting the inner products by
definitions of Uµ and fµ(v), we get
〈ω̃Ak(p), v〉 = (fµ)k(v)−
dµ∑
l=k+1
(Uµ)k,l〈ω̃Al(p), v〉. (6.3)
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 17
One may recognize this as the backward substitution formula for calculating linear equations of
the form Ax = b, where A is a upper unitriangular matrix. Whence,
v =
dµ∑
k=1
(
U−1µ · fµ(v)
)
k
ω̃Ak(p).
Repeated use of (6.3) shows that
v =
dµ∑
k=1
dµ−k∑
t=0
∑
k=l0<···<lt≤dµ
(−1)t
t∏
s=1
(Uµ)ls−1,ls(fµ)ltω̃Ak(p). �
The following is a sketch of how to determine the expansion of Xiω̃A(p) using Proposition 6.2.
We do not include calculations of the entries of Uµ and fµ(Xiω̃A(p)). Consider the case n = 4,
p = 2, i = 1 and
A =
2 3
4
Then X1ω̃ 2 3
4
(2) is a linear combination of the vectors ω̃B(2), for B ∈ Y4(2) of weight µ =
µA + ε1 = (1, 1, 1, 1). Since dµ = 6, there are 6 s.s. Young tableaux of weight (1, 1, 1, 1). These
are, in increasing order,
A1 =
1 3
2 4
, A2 =
1 3 4
2
, A3 =
1 2
3 4
, A4 =
1 2 4
3
,
A5 =
1 2 3
4
, A6 = 1 2 3 4 .
Had we instead chosen p ≥ 4, then we would also have to take into account the 4 remaining
s.s. Young tableaux of weight (1, 1, 1, 1):
1
2
3
4
,
1 4
2
3
,
1 3
2
4
and
1 4
2
3
. When p = 2 no basis vectors
correspond to these tableaux as they have strictly more than 2 rows.
Calculating the coefficients of the relevant terms we get
Uµ =
1 −1 0 0 1 1
0 1 0 0 0 −1
0 0 1 −1 −1 −1
0 0 0 1 0 1
0 0 0 0 1 −1
0 0 0 0 0 1
and fµ
(
X1ω̃ 2 3
4
(2)
)
=
0
−1
1
0
1
0
.
Note that in general the entries of Uµ take values in all of Z and not just in {±1, 0}. Calculating
the inverse of Uµ we get
U−1µ =
1 1 0 0 −1 −1
0 1 0 0 0 1
0 0 1 1 1 1
0 0 0 1 0 −1
0 0 0 0 1 1
0 0 0 0 0 1
18 A.K. Bisbo, H. de Bie and J. Van der Jeugt
and
U−1µ · fµ
(
X1ω̃ 2 3
4
(2)
)
= (−2,−1, 2, 0, 1, 0).
Using Proposition 6.2 this tells us that
X1ω̃ 2 3
4
(2) = −2ω̃ 1 3
2 4
(2)− ω̃ 1 3 4
2
(2) + 2ω̃ 1 2
3 4
(2) + ω̃ 1 2 3
4
(2).
To determine general formulas for the entries of Uµ, fµ(Xiω̃A(p)) and fµ(Diω̃A(p)) we first
write them as functions of the coefficients cB(γ) introduced right before Lemma 5.4. Simple
calculations and applications of definitions yield the following formulas.
Lemma 6.3. Let A,B ∈ Yn(p) and i ∈ {1, . . . , n}. Then
〈xB,DBeDB , ω̃A(p)〉 = cB(γB,DB )cA(γB,DB ),
〈xB,DBeDB , Xiω̃A(p)〉 = cB(γB,DB )
p∑
α=1
α−1∏
β=1
(−1)(λB)βcA(γB,DB − εi,α),
〈xB,DBeDB , Diω̃A(p)〉 = cB(γB,DB )
p∑
α=1
α−1∏
β=1
(−1)(λB)β ((γB,DB )i,α + 1)cA(γB,DB + εi,α),
where cA(γB,DB − εi,α) := 0 if (γB,DB )i,α = 0, and where εi,α ∈Mn,p(N0) is the matrix with
(εi,α)j,β := δi,jδα,β,
for all i, j ∈ {1, . . . , n} and α, β ∈ {1, . . . , p}.
Lemma 5.4 already gives a formula for cB(γB,DB ). The rest of this section will be dedicated to
obtaining concrete formulas for the calculation of cA(γ) given any A ∈ Yn(p) and γ ∈Mn,p(N0).
Consider first the expression given in Lemma 5.4
cA(γ) =
∑
C∈T(λA,p),
γ=γA,C
(−1)N(C),
where N(C) ∈ N0 is a number such that
xA,CeC = (−1)N(C)xγA,CeηγA,C .
The general idea behind our approach to the calculation of cA(γ) is to determine the set
{C ∈ T(λA, p) : γA,C = γ},
and for each element in this set calculate the number (−1)N(C). Consider the following more
general class of Young tableaux,
E(λ, p) := {fillings of the Young diagram of shape λ ∈ P by numbers 1, . . . , p}.
We shall refer to the elements of E(λ, p) as Young tableaux. Note that T(λ, p) ⊂ E(λ, p). The
definitions (5.1) and (5.2) can naturally be extended to hold, for any A ∈ Yn(p) with λ = λA
and T ∈ E(λ, p), by letting m = (λA)1 and defining
eT :=
(
et1,m · · · etλ′m,m
)
· · ·
(
et1,1 · · · etλ′1,1
)
∈ C`p
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 19
and
xA,T :=
(
xa1,m,t1,m · · ·xaλ′m,m,tλ′m,m
)
· · ·
(
xa1,1,t1,1 · · ·xaλ′1,1,tλ′1,1
)
∈ C[Rnp].
Let furthermore γA,T ∈ Mn,p(N0) and N(T ) such that xA,T eT = (−1)N(T )xγA,T eηγA,T . Define
now, for each µ ∈ Nn0 , the following permutation group,
Sµ := Sµ1 × · · · × Sµn .
The reason for introducing these objects is that the set E(λ, p) carries a useful action of the
permutation group Sµ, for each A ∈ Yn(p) with µA = µ and λA = λ. To define this action, let
yA(i, s) = (kA(i, s), lA(i, s)) ∈ λA,
for i ∈ {1, . . . , n} and s ∈ {1, . . . , (µA)i}, be the coordinates of λA such that
λAi − λAi−1 =
{
yA(i, s) : s ∈ {1, . . . , (µA)i}
}
,
and lA(i, 1) > · · · > lA(i, (µA)i). Here λAi − λAi−1 is the set of coordinates y ∈ λA whose
corresponding entries ay in A are i. Specifically
lA(i, s) = max
{
l ∈ {1, . . . , (λAi)1} : s =
(λAi )1∑
r=l
(λAi)
′
r − (λAi−1)′r
}
,
kA(i, s) = (λAi)
′
lA(i,s)
.
Note that these coordinates cover all entries of λA
λA =
{
yA(i, s) : i ∈ {1, . . . , n}, s ∈ {1, . . . , (µA)i}
}
.
In the terminology of Macdonald [24], {yA(i, s) : s ∈ {1, . . . , (µA)i}} is the i’th horizontal strip
of A consisting of the boxes i and indexed from right to left by s. As an example consider
A =
1 1 1 2 2
2 2 3
3
Then µ1 = 3, µ2 = 4 and µ3 = 2. For i = 2, the coordinates yA(2, s) ∈ λA for s ∈ {1, 2, 3, 4} are
given by
yA(2, 1) = (1, 5), yA(2, 2) = (1, 4), yA(2, 3) = (2, 2) and yA(2, 4) = (2, 1).
Visually these coordinates refer to the gray boxes in the Young diagram λA:
The action
πA : SµA × E(λA, p)→ E(λA, p),
is defined by letting T σ,A := πA(σ, T ) be the Young tableau with entries tσ,Ay , for y ∈ λA,
defined by
tσ,AyA(i,s) := tyA(i,σ−1
i (s)), (6.4)
20 A.K. Bisbo, H. de Bie and J. Van der Jeugt
for all i ∈ {1, . . . , n} and s ∈ {1, . . . , (µA)i}. Consider A as in the previous example, and let
p = 4 and
T =
2 1 1 2 2
4 3 3
3
∈ T((5, 3, 1), 4).
Let I denote the identity permutation, and let σ, τ ∈ SµA = S3 × S4 × S2 with
σ = (I, (13), I) and τ = (I, (2413), I).
The πA action of these permutations only permutes the entries of T corresponding to the coor-
dinates of the 2nd horizontal strip of A. The boxes corresponding to these coordinates are
marked with gray below:
T σ,A =
2 1 1 2 3
4 2 3
3
and T τ,A =
2 1 1 3 4
2 2 3
3
Note here that T σ,A ∈ T((5, 3, 1), 4), whereas T τ,A /∈ T((5, 3, 1), 4).
Lemma 6.4. Let A ∈ Yn(p) and T, T ′ ∈ E(λA, p), then xA,T = xA,T ′ if and only if there exists
σ ∈ SµA such that T ′ = T σ,A.
Equivalently this lemma states that{
T ′ ∈ E(λA, p) : γA,T ′ = γA,T
}
=
{
T σ,A : σ ∈ SµA
}
.
Proof. Let ay, ty and t′y, for y ∈ λA, denote the entries of A, T and T ′ respectively. Note first
that, for all σ = (σ1, . . . , σn) ∈ SµA ,
xA,T =
n∏
i=1
(µA)i∏
s=1
xayA(i,s),tyA(i,s)
=
n∏
i=1
(µA)i∏
s=1
xayA(i,s),tyA(i,σ−1
i
(s))
= xA,Tσ,A .
This yields one of the implications stated in the lemma. If xA,C = xA,C′ then, for all i ∈
{1, . . . , n},
(µA)i∏
s=1
xayA(i,s),tyA(i,s)
=
(µA)i∏
s=1
xayA(i,s),t
′
yA(i,s)
.
This implies that the vectors
(
tyA(i,1), . . . , tyA(i,(µA)i)
)
and
(
t′yA(i,1), . . . , t
′
yA(i,(µA)i)
)
contain the
same amount of α-entries, for all α ∈ {1, . . . , p}, letting us construct a permutation σi ∈ S(µA)i
such that
t′yA(i,s) = tyA(i,σ−1
i (s)),
for all s ∈ {1, . . . , (µA)i}. Repeating this process for all i ∈ {1, . . . , n} and defining σ :=
(σ1, . . . , σn), it is clear that σ ∈ SµA and T ′ = T σ,A. �
A Young tableau Tγ,A ∈ E(λA, p) for which γA,Tγ,A = γ can be constructed in the following
way. For all α ∈ {1, . . . , p}, i ∈ {1, . . . , n} and s with
∑α−1
β=1 γi,β < s ≤
∑α
β=1 γi,β we define the
entry of Tγ,A at the coordinate yA(i, s) to be
(tγ,A)yA(i,s) := α. (6.5)
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 21
That is, for any given i ∈ {1, . . . , n} we let the boxes of the i’th horizontal strip be filled, from
right to left, by γi,1 1’s, then γi,2 2’s, followed by γi,3 3’s and so on.
As an example consider n = 3, p = 4,
A =
1 1 1 2 2
2 2 3
3
and γ =
2 1 0 0
0 2 1 1
0 0 2 0
. (6.6)
In that case
Tγ,A =
2 1 1 2 2
4 3 3
3
The gray boxes correspond to the 2nd horizontal strip. A notable feature of this construction is
that TγA,DA ,A = DA.
By applying Lemma 6.4 we can now calculate cA(γ). This is the content of Proposition 6.5.
For any A ∈ Yn(p) and γ ∈Mn(p) with µγ = µA, let
fγ,A(σ) :=
{
(−1)N(Tσ,Aγ,A ), if T σ,Aγ,A ∈ T(λA, p),
0, otherwise,
(6.7)
for all σ ∈ SµA . Here T σ,Aγ,A is the permutation of Tγ,A, defined in (6.5), by σ via the action
defined in (6.4).
Proposition 6.5. For any A ∈ Yn(p) and γ ∈Mn(p) with µγ = µA,
cA(γ) =
∏
1≤i≤n
1≤α≤p
1
γi,α!
∑
σ∈SµA
fγ,A(σ). (6.8)
In (6.8) the constant preceding the sum can be considered an over-counting factor since∏
1≤i≤n
1≤α≤p
γi,α! = #
{
σ′ ∈ SµA : T σ
′,A
γ,A = T σ,Aγ,A
}
. (6.9)
To obtain a simple formula for the signs (−1)N(T ) we consider the following total order on
the coordinates (k, l) of λ ∈ P
(k, l) < (k′, l′) if and only if l > l′, or l = l′ and k < k′,
that is
(1, λ1) < · · · < (λ′λ1 , λ1) < · · · < (1, 1) < · · · < (λ′1, 1).
This ordering is motivated by the definition
eT = et(1,λ1) · · · et(λ′λ1 ,λ1)
· · · et(1,1) · · · et(λ′1,1) ,
for all T ∈ E(λ, p). With it we can write
eT = (−1)#{(k,l),(k
′,l′)∈λ : (k,l)>(k′,l′) and t(k,l)<t(k′,l′)}e
(ηγA,T )1
1 · · · e
(ηγA,T )p
p ,
22 A.K. Bisbo, H. de Bie and J. Van der Jeugt
which implies that
(−1)N(T ) = (−1)#{(k,l),(k
′,l′)∈λ : (k,l)>(k′,l′) and t(k,l)<t(k′,l′)}. (6.10)
We have now reached a point where any coefficient of the form cA(γ) can be explicitly
calculated. To show how such a calculation is done consider A and γ as in (6.6), with n = 3
and p = 4. Here µA = (3, 4, 2) meaning that SµA contains |SµA | = 3!4!2! = 288 different
permutations. Taking the over-counting factor (6.9) into account the total number of Young
tableaux generated by the permutation action πA is
#
{
T σ,Aγ,A : σ ∈ SµA
}
=
∏
1≤i≤n
1≤α≤p
|SµA |
γi,α!
=
3!4!2!
23
= 36.
Out of these 36 tableaux 19 are not column distinct and thus contribute with factors of 0 to cA(γ),
see (6.7). An example of this is obtained by letting τ = (I, (2413), I). The corresponding tableau
T τ,Aγ,A =
2 1 1 3 4
2 2 3
3
has two entries containing a 2 in its first column and is thus not column distinct. The remaining
17 tableaux are column distinct and each contribute with a factor of ±1 to cA(γ). Examples
from these 17 tableaux include the ones corresponding to permutations
σ = (I, (13), I) and κ = ((13), (2413), I).
Those are
T σ,Aγ,A =
2 1 1 2 3
4 2 3
3
and T κ,Aγ,A =
1 1 2 3 4
2 2 3
3
Calculated with (6.10) the contributions of these tableaux are
(−1)N(Tσ,Aγ,A ) = (−1)11 = −1 and (−1)N(Tκ,Aγ,A ) = (−1)20 = 1.
Out of all 17 column distinct tableaux one will find that 9 have positive sign and 8 have negative
meaning that
cA(γ) = 9− 8 = 1.
Obtaining the coefficients cA(γ) in this way requires the construction of all the relevant Young
tableaux. Formulas for doing these calculations without explicitly constructing the tableaux
become rather complicated. For the interested readers such formulas are included in Appendix B.
In this section, we have so far explained how to calculate all the coefficients in the expansions
from (6.2). Such calculations are done by combining Proposition 6.2 with Proposition 6.5 and
Lemma 6.3. Proposition 6.6 shows how, using the coefficients from (6.2), we can obtain the
expansions for the actions of B+
i and B−i on the basis vectors ṽA(p) = 1
λA!
vA(p) of V n(p).
Proposition 6.6. Let i ∈ {1, . . . , n} and A,B ∈ Yn, then
B+
i ṽA(p) =
∑
B∈Yn
c̄B(i, n,A)ṽB(p), (6.11)
B−i ṽA(p) =
∑
B∈Yn
(
(n+ 1− p)ĉB(i, n,A)− (n− p)ĉB(i, n+ 1, A)
)
ṽB(p). (6.12)
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 23
By Theorem 5.7, formulas corresponding to (6.12) also hold for the actions of Xi and Di
on Ln(p), though with the sums only being over B ∈ Yn(p). Proposition 6.6 tells us that
by using the tools presented in this section to determine matrix coefficients of the Xi and Di
actions on Ln(n) and Ln(n + 1) we can determine the matrix coefficients of any other action,
be it on Ln(p) or V n(p), by simple linear combination of coefficients.
Proof. Let |0〉 be the lowest weight vector of V n(p). Corollary 5.8 tells us that there exist
coefficients dB(i, A), independent of p, such that
B+
i ṽA(p) = B+
i
(
1
λA!
B+
A
)
|0〉 =
∑
B∈Yn
dB(i, A)
1
λB!
B+
B |0〉 =
∑
B∈Yn
dB(i, A)ṽB(p).
Since Ψn is an isomorphism of vector spaces with Ψn(ω̃B(n)) = ṽB(n), it follows that dB(i, A) =
c̄B(i, n,A), for all B ∈ Yn = Yn(n), when we compare with the coefficients in (6.2).
Using (2.1) we can find B1,+(i, j, A), B2,+(i, A), B3,+(i, A) ∈ U(osp(1|2n)+) such that
B−i B
+
A =
( ∑
1≤i≤j≤n
B1,+(i, j, A)
{
B−i , B
+
j
})
+B2,+(i, A) +B3,+(i, A)B−i .
Corollary 5.8 together with (2.3) then tells us that there exists coefficients d1B(i, A) and d2B(i, A),
independent of p, such that
B−i ṽA(p) = B−i
(
1
λA!
B+
A
)
|0〉 (6.13)
=
∑
B∈Yn
d1B(i, A)
1
λB!
B+
B{B
−
i , B
+
i }|0〉+
∑
B∈Yn
d2B(i, A)
1
λB!
B+
B |0〉 (6.14)
=
∑
B∈Yn
(
d1B(i, A)p+ d2B(i, A)
)
ṽB(p). (6.15)
The fact that Ψn and Ψn+1 are isomorphisms of vector spaces with Ψn(ω̃B(n)) = ṽB(n)
and Ψn+1(ω̃B(n+1)) = ṽB(n+1) lets us compare coefficients with (6.2) and obtain the equations
d1B(i, A)n+ d2B(i, A) = ĉB(i, n,A),
and
d1B(i, A)(n+ 1) + d2B(i, A) = ĉB(i, n+ 1, A).
Solving these equations for d1B(i, A) and d2B(i, A) and inserting the results into (6.15) gives the
statement of the proposition. �
7 Example: the case n = 2
In the simplest non-trivial case, that is when n = 2, the action of osp(1|4) on the tableau vectors
is particularly simple. The illustration of this is the subject of this short section. For any
k, l,m ∈ N0 we let A(k, l,m) denote the following s.s. Young tableau in Y2(p)
A(k, l,m) := 1
2
· · · 1
2︸ ︷︷ ︸
m
1 · · · 1︸ ︷︷ ︸
l
2 · · · 2︸ ︷︷ ︸
k
.
24 A.K. Bisbo, H. de Bie and J. Van der Jeugt
Any s.s. Young tableau from Y2(p) can be written in this form. We note that in this descriptionm
refers to the number of 1
2
-columns in A(k, l,m) and similarly that l and k denote the number
of 1 - and 2 -columns in A(k, l,m) respectively. The corresponding tableau vector is then
ωA(k,l,m)(p) = Xk
2X
l
1[X1, X2]
m.
The actions of the generators X1, X2, D1 and D2 of osp(1|4) on ωA(k,l,m)(p) are then given as
follows
X1ωA(k,l,m)(p) = ωA(k,l+1,m)(p) + (−1)l[k]2ωA(k−1,l,m+1)(p), (7.1)
X2ωA(k,l,m)(p) = ωA(k+1,l,m)(p), (7.2)
D1ωA(k,l,m)(p) = (−1)k+l(2m+ (2p− 4)[m]2)ωA(k+1,l,m−1)(p) (7.3)
+
(
l + (−1)k
(
(−1)mp− 1 + 4[m]2
)
[l]2
)
ωA(k,l−1,m)(p) (7.4)
+ (−1)l[k]2(l − [l]2)ωA(k−1,l−2,m+1)(p), (7.5)
D2ωA(k,l,m)(p) = (−1)k+l+1(2m+ (2p− 4)[m]2)ωA(k,l+1,m−1)(p) (7.6)
+
(
k + (2m+ p− 1)[k]2
)
ωA(k−1,l,m)(p), (7.7)
where [k]2 = 0 if k is even and [k]2 = 1 if k is odd.
The actions in (7.7) hold for any p ∈ N. When p = 1 we have [X1, X2] = 0, meaning that the
tableau vectors reduce to regular monomials when we disregard the, in this case, trivial Clifford
algebra part. That is, ωA(k,l,0)(1) = xk2,1x
l
1,1 and ωA(k,l,m)(1) = 0, for all k, l ∈ N0 and m > 0.
The actions of the osp(1|4) generators similarly reduce to those of usual variable multiplication
and differentiation.
A Graded lexicographic order
The definitions of the graded lexicographic ordering on Nn0 and P were omitted in the main text.
For the readers unfamiliar with these orderings the definitions are included here.
Definition A.1. Given µ, ν ∈ Nn0 we say that µ < ν with respect to the graded lexicographic
ordering if |µ| =
∑n
i=1 µi < |ν| =
∑n
i=1 νi, or if |µ| = |ν| and µj < νj for the first j, where µj
and νj differ.
For n = 4, the µ ∈ N4
0 with |µ| ≤ 2 are ordered as follows,
(0, 0, 0, 0) < (0, 0, 0, 1) < (0, 0, 1, 0) < (0, 1, 0, 0) < (1, 0, 0, 0) < (0, 0, 0, 2)
< (0, 0, 1, 1) < (0, 0, 2, 0) < (0, 1, 0, 1) < (0, 1, 1, 0) < (0, 2, 0, 0)
< (1, 0, 0, 1) < (1, 0, 1, 0) < (1, 1, 0, 0) < (2, 0, 0, 0).
Definition A.2. Given λ, κ ∈ P we say that λ < κ with respect to the graded lexicographic
ordering if |λ| < |κ|, or if |λ| = |κ| and λj < κj for the first j, where λj and κj differ.
The λ ∈ P ∈ Nn0 with |λ| ≤ 4 are ordered as follows,
(0)< (1)< (1, 1)< (2)< (1, 1, 1)< (2, 1)< (3)< (1, 1, 1, 1)< (2, 1, 1)< (2, 2)< (3, 1)< (4).
Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases 25
B Alternative formula for cA(γ)
We present here a formula for calculating the coefficients cA(γ) of ω̃A(p) that is more explicit
than the one produced at the end of Section 6. What follows are two technical lemmas in
which constituents of the final formula are obtained. Following those lemmas the final formula
is presented, marking the end of this appendix. Due to them being technical and not very
enlightening the proofs of the following results are omitted.
Consider the function
G(l, k, l′) =
l′ − l, if l ≤ l′ ≤ k,
l′ − k, if l ≤ k ≤ l′,
0, otherwise,
for all l, k, l′ ∈ N0.
Lemma B.1. Given A ∈ Yn(p) and γ ∈Mn(p) with µγ = µA, then
eTγ,A = (−1)N1(γ,A)+N2(γ,A)eηγ ,
where
N1(γ,A) =
∑
1≤α<α′≤p,
1≤i<i′≤n
∑α
β=1 γi,β∑
s=1+
∑α−1
β=1 γi,β
G
(
α′−1∑
β=1
γi′,β,
(λ
Ai
′ )1∑
l=lA(i,s)+1
(λAi′ )
′
l − (λAi′−1)′l,
α′∑
β=1
γi′,β
)
and
N2(γ,A) =
∑
1≤α<α′≤p,
1≤i′<i≤n
∑α
β=1 γi,β∑
s=1+
∑α−1
β=1 γi,β
G
(
α′−1∑
β=1
γi′,β,
(λ
Ai
′ )1∑
l=lA(i,s)
(λAi′ )
′
l − (λAi′−1)′l,
α′∑
β=1
γi′,β
)
.
Let k ∈ N and consider a k-tuple of positive integers L = (L1, . . . , Lk) ∈ Nk. If Lm 6= Lm′ ,
for all m 6= m′, then we define σL ∈ Sk to be the permutation such that
Lσ(1) < · · · < Lσ(k).
With this we can define the sign of L to be
sgn(L) =
{
sgn(σL), if Lm 6= Lm′ , for all m 6= m′,
0, otherwise.
Lemma B.2. Given A ∈ Yn(p), σ ∈ SµA and γ ∈Mn,p(N0) with µγ = µA, then
sgn(σ)
p∏
α=1
sgn(Lγ,A(σ, α))(−1)N0(γ,A)eTγ,A =
{
e
Tσ,Aγ,A
, if T σ,Aγ,A ∈ T(λA, p),
0, otherwise,
where
N0(γ,A) =
∑
1≤α≤p,
1≤i<i′≤n
∑α
β=1 γi,β∑
s=1+
∑α−1
β=1 γi,β
G
(
α−1∑
β=1
γi′,β,
(λ
Ai
′ )1∑
l=lA(i,s)+1
(λAi′ )
′
l − (λAi′−1)′l,
α∑
β=1
γi′,β
)
26 A.K. Bisbo, H. de Bie and J. Van der Jeugt
and Lγ,A(σ, α) ∈ N(ηγ)α defined such that, for all α ∈ {1, . . . , p}, i ∈ {1, . . . , n} and t with∑i−1
j=1 γj,α < t ≤
∑i
j=1 γj,α,
(
Lγ,A(σ, α)
)
t
= lA
(
i, σ−1i
(
t−
i−1∑
j=1
γj,α +
α−1∑
β=1
γi,β
))
.
If (ηγ)α = 0, then sgn(Lγ,A(σ, α)) := 1.
Proposition B.3. Let A ∈ Yn(p) and γ ∈Mn,p(N0) with µγ = µA, then
cA(γ) =
∑
σ∈SµA
∏
1≤i≤n
1≤α≤p
1
γi,α!
sgn(σ)(−1)N(γ,A) sgn(LA(σ, α)),
where N(γ,A) = N0(γ,A) +N1(γ,A) +N2(γ,A).
Acknowledgements
The authors were supported by the EOS Research Project 30889451. The editor and referees
are thanked for their helpful reports.
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1 Introduction
2 The polynomial paraboson Fock space Ln(p)
3 The induced module Vn(p)
4 Tableau vectors in Ln(p) and Vn(p)
5 Bases for Ln(p) and Vn(p)
6 Action of osp(1|2n) on tableau vectors omega A(p)
7 Example: the case n=2
A Graded lexicographic order
B Alternative formula for cA(gamma)
References
|
| id | nasplib_isofts_kiev_ua-123456789-211318 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T17:41:18Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bisbo, Asmus K. De Bie, Hendrik Van der Jeugt, Joris 2025-12-29T11:10:16Z 2021 Representations of the Lie Superalgebra (1|2) with Polynomial Bases. Asmus K. Bisbo, Hendrik De Bie and Joris Van der Jeugt. SIGMA 17 (2021), 031, 27 pages 1815-0659 2020 Mathematics Subject Classification: 17B10; 05E10; 81R05; 15A66 arXiv:1912.06488 https://nasplib.isofts.kiev.ua/handle/123456789/211318 https://doi.org/10.3842/SIGMA.2021.031 We study a particular class of infinite-dimensional representations of (1|2). These representations ₙ() are characterized by a positive integer p, and are the lowest component in the p-fold tensor product of the metaplectic representation of (1|2). We construct a new polynomial basis for ₙ() arising from the embedding (1|2) ⊃ (1|2). The basis vectors of ₙ() are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra-valued polynomials with integer coefficients in variables. Using combinatorial properties of these tableau vectors, it is deduced that they form a basis. The computation of matrix elements of a set of generators of (1|2) on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau. The authors were supported by the EOS Research Project 30889451. The editor and referees are thanked for their helpful reports. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Representations of the Lie Superalgebra (1|2n) with Polynomial Bases Article published earlier |
| spellingShingle | Representations of the Lie Superalgebra (1|2n) with Polynomial Bases Bisbo, Asmus K. De Bie, Hendrik Van der Jeugt, Joris |
| title | Representations of the Lie Superalgebra (1|2n) with Polynomial Bases |
| title_full | Representations of the Lie Superalgebra (1|2n) with Polynomial Bases |
| title_fullStr | Representations of the Lie Superalgebra (1|2n) with Polynomial Bases |
| title_full_unstemmed | Representations of the Lie Superalgebra (1|2n) with Polynomial Bases |
| title_short | Representations of the Lie Superalgebra (1|2n) with Polynomial Bases |
| title_sort | representations of the lie superalgebra (1|2n) with polynomial bases |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211318 |
| work_keys_str_mv | AT bisboasmusk representationsoftheliesuperalgebra12nwithpolynomialbases AT debiehendrik representationsoftheliesuperalgebra12nwithpolynomialbases AT vanderjeugtjoris representationsoftheliesuperalgebra12nwithpolynomialbases |