Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups
A Young subgroup of the symmetric group N, the permutation group of {1, 2, …, }, is generated by a subset of the adjacent transpositions {(, +1)∣1 ≤ < }. Such a group is realized as the stabilizer ₙ of a monomial λ (=λ¹₁λ²₂ ⋯ λᴺN) with λ = (ⁿ¹₁, ⁿ²₂, …, ⁿᵖₚ) (meaning ⱼ is repeated ⱼ times, 1...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
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Інститут математики НАН України
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| Цитувати: | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups. Charles F. Dunkl. SIGMA 21 (2025), 053, 17 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860295037122772992 |
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| author | Dunkl, Charles F. |
| author_facet | Dunkl, Charles F. |
| citation_txt | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups. Charles F. Dunkl. SIGMA 21 (2025), 053, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A Young subgroup of the symmetric group N, the permutation group of {1, 2, …, }, is generated by a subset of the adjacent transpositions {(, +1)∣1 ≤ < }. Such a group is realized as the stabilizer ₙ of a monomial λ (=λ¹₁λ²₂ ⋯ λᴺN) with λ = (ⁿ¹₁, ⁿ²₂, …, ⁿᵖₚ) (meaning ⱼ is repeated ⱼ times, 1 ≤ ≤ , and ₁ > ₂ > ⋯ > ₚ ≥ 0), thus it is isomorphic to the direct product ₙ₁ × ₙ₂ × ⋯ ×ₙₚ. The interval {1, 2, …, } is a union of disjoint sets ⱼ = { ∣ λᵢ = ⱼ}. The orbit of λ under the action of N (by permutation of coordinates) spans a module λ, the representation induced from the identity representation of ₙ. The space λ decomposes into a direct sum of irreducible N-modules. The spherical function is defined for each of these; it is the character of the module averaged over the group ₙ. This paper concerns the value of certain spherical functions evaluated at a cycle that has no more than one entry in each interval ⱼ. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by . Gorin and the author [arXiv:2412:01938]. In particular, the present paper determines the spherical function value for N-modules of hook tableau type, corresponding to Young tableaux of shape [ − , 1ᵇ].
|
| first_indexed | 2026-03-21T09:33:16Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 053, 17 pages
Some Spherical Function Values
for Hook Tableaux Isotypes and Young Subgroups
Charles F. DUNKL
Department of Mathematics, University of Virginia,
PO Box 400137, Charlottesville VA 22904-4137, USA
E-mail: cfd5z@virginia.edu
URL: https://uva.theopenscholar.com/charles-dunkl
Received March 12, 2025, in final form June 30, 2025; Published online July 08, 2025
https://doi.org/10.3842/SIGMA.2025.053
Abstract. A Young subgroup of the symmetric group SN , the permutation group of
{1, 2, . . . , N}, is generated by a subset of the adjacent transpositions {(i, i+1) | 1 ≤ i < N}.
Such a group is realized as the stabilizer Gn of a monomial xλ
(
=xλ1
1 xλ2
2 · · ·xλN
N
)
with
λ =
(
dn1
1 , dn2
2 , . . . , d
np
p
)
(meaning dj is repeated nj times, 1 ≤ j ≤ p, and d1 > d2 > · · · >
dp ≥ 0), thus is isomorphic to the direct product Sn1
× Sn2
× · · · × Snp
. The interval
{1, 2, . . . , N} is a union of disjoint sets Ij = {i | λi = dj}. The orbit of xλ under the action
of SN (by permutation of coordinates) spans a module Vλ, the representation induced from
the identity representation of Gn. The space Vλ decomposes into a direct sum of irreducible
SN -modules. The spherical function is defined for each of these, it is the character of the
module averaged over the group Gn. This paper concerns the value of certain spherical
functions evaluated at a cycle which has no more than one entry in each interval Ij . These
values appear in the study of eigenvalues of the Heckman–Polychronakos operators in the
paper by V. Gorin and the author [arXiv:2412:01938]. In particular, the present paper de-
termines the spherical function value for SN -modules of hook tableau type, corresponding
to Young tableaux of shape
[
N − b, 1b
]
.
Key words: spherical functions; subgroups of the symmetric group; hook tableaux; alternat-
ing polynomials
2020 Mathematics Subject Classification: 20C30; 43A90; 20B30
To the memory of G. de B. Robinson (1906–1992)
my first algebra professor
1 Introduction
There is a commutative family of differential-difference operators acting on polynomials in N
variables whose symmetric eigenfunctions are Jack polynomials. They are called Heckman–
Polychronakos operators, defined by Pk :=
∑N
i=1(xiDi)
k, k = 1, 2, . . . , in terms of Dunkl opera-
tors
Dif(x) :=
∂
∂xi
f(x) + κ
N∑
j=1,j ̸=i
f(x)− f(x(i, j))
xi − xj
;
x(i, j) denotes x with xi and xj interchanged, and κ is a fixed parameter, often satisfying κ > − 1
N
(see Heckman [4], Polychronakos [6]; these citations motivated the name given the operators
in [3]). The symmetric group on N objects, that is, the permutation group of {1, 2, . . . , N},
is denoted by SN and acts on R[x1, . . . , xN ] by permutation of the variables. Specifically, for
a polynomial f(x) and w ∈ SN the action is wf(x) = f(xw), (xw)i = xw(i), 1 ≤ i ≤ N . This
is a representation of SN . The operators Pk commute with this action and thus the structure
mailto:cfd5z@virginia.edu
https://uva.theopenscholar.com/charles-dunkl
https://doi.org/10.3842/SIGMA.2025.053
2 C.F. Dunkl
of eigenfunctions and eigenvalues is strongly connected to the decomposition of the space of
polynomials into irreducible SN -modules. The latter are indexed by partitions of N , that is,
τ = (τ1, . . . , τℓ) with τi ∈ N, τ1 ≥ τ2 ≥ · · · ≥ τℓ > 0 and
∑ℓ
i=1 τi = N . The corresponding module
is spanned by the standard Young tableaux of shape τ . The general details are not needed here.
The types of polynomial modules of interest here are spans of certain monomials: for α ∈ ZN+ , let
xα :=
∏N
i=1 x
αi
i . Suppose λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0, then set Vλ = spanF
{
xβ | β = wλ, w ∈ SN
}
,
that is, β ranges over the permutations of λ, and F is an extension field of R containing at
least κ. The space Vλ is invariant under the action of SN . The eigenvector analysis of Pk is
based on the triangular decomposition PkVλ ⊂ Vλ ⊕
∑
ν≺λ⊕Vν , where ν ≺ λ is the dominance
order,
∑j
i=1 νi ≤
∑j
i=1 λi for 1 ≤ j ≤ N , and
∑N
i=1 νi =
∑N
i=1 λi. Part of the analysis is to
identify irreducible SN -submodules of Vλ. This depends on the number of repetitions of values
among {λi | 1 ≤ i ≤ N}. To be precise let λ =
(
dn1
1 , d
n2
2 , . . . , d
np
p
)
(that is, dj is repeated
nj times, 1 ≤ j ≤ p), with d1 > d2 > · · · > dp ≥ 0 and N =
∑p
i=1 ni. Let Gn denote the
stabilizer group of xλ, so that Gn
∼= Sn1 × Sn2 × · · · × Snp . The representation of SN realized
on Vλ is the induced representation indSN
Gn
. This decomposes into irreducible SN -modules and
the number of copies (the multiplicity) of a particular isotype τ in Vλ is called a Kostka number
(see Macdonald [5, p. 101]).
The operator Pk arose in the study of the Calogero–Sutherland quantum system ofN identical
particles on a circle with inverse-square distance potential: the Hamiltonian is
H = −
N∑
j=1
(
∂
∂θj
)2
+
κ(κ− 1)
2
∑
1≤i<j≤N
1
sin2
(
1
2(θi − θj)
) ;
the particles are at θ1, . . . , θN and the chordal distance between two points is
∣∣2 sin(12(θi− θj))∣∣.
By changing variables xj = exp iθj , the Hamiltonian is transformed to
H =
N∑
j=1
(
xj
∂
∂xj
)2
− 2κ(κ− 1)
∑
1≤i<j≤N
xixj
(xi − xj)2
(for more details see Chalykh [2, p. 16]).
In [3], Gorin and the author studied the eigenvalues of the operators Pk restricted to sub-
modules of Vλ of given isotype τ . It turned out that if the multiplicity of the isotype τ in Vλ
is greater than one, then the eigenvalues are not rational in the parameters and do not seem
to allow explicit formulation. However, the sum of all the eigenvalues (for any fixed k) can
be explicitly found, in terms of the character of τ . In general, this may not have a relatively
simple form but there are cases allowing a closed form. The present paper carries this out for
hook isotypes, labeled by partitions of the form
[
N − b, 1b
]
(the Young diagram has a hook
shape). The formula for the sum is quite complicated with a number of ingredients. For given
(n1, . . . , np) define the intervals associated with λ, Ij =
[∑j−1
i=1 ni + 1,
∑j
i=1 ni
]
for 1 ≤ j ≤ p
(notation [a, b] := {a, a + 1, . . . , b} ⊂ N). The formula is based on considering cycles gA corre-
sponding to subsets A ={a1, . . . , aℓ} of [1, p], which are of length ℓ with exactly one entry from
each interval Iaj . Any such cycle can be used and the order of a1, . . . , aℓ does not matter. The
degrees d1, . . . , dp enter the formula in a shifted way:
d̃i := di + κ(ni+1 + ni+2 + · · ·+ np), 1 ≤ i ≤ p.
Let hAm := hm
(
d̃a1 , d̃a2 , . . . , d̃aℓ
)
, the complete symmetric polynomial of degree m (the gener-
ating function is
∑
k≥0 hk(c1, c2, . . . , cq)t
k =
∏q
i=1(1 − cit)
−1, see [5, p. 21]). In [3], we used
an “averaged character” (spherical function, in the present paper). Denote the character of the
representation τ of SN by χτ , then
χτ [A;n] :=
1
#Gn
∑
h∈Gn
χτ (gAh),
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 3
where gA is an ℓ-cycle labeled by A as above, and #Gn =
∏p
i=1 ni!. In formula (1.1), the inner
sum is over k-subsets of [1, p], the list of labels of the partition {I1, . . . , Ip}, so a typical subset is
{a1, a2, . . . , ak}, gA is a cycle with one entry in each Iai and h
A
k+1−ℓ = hk+1−ℓ
(
d̃a1 , d̃a2 , . . . , d̃aℓ
)
.
Now suppose the multiplicity of τ in Vλ is µ, then there are µdim τ eigenfunctions and
eigenvalues of Pk, and the sum of all these eigenvalues is [3, Theorem 5.4]
dim τ
min(k+1,p)∑
ℓ=1
(−κ)ℓ−1
∑
A⊂[1,p],#A=ℓ
χτ [A;n]hAk+1−ℓ
∏
i∈A
ni!. (1.1)
The main result of the present paper is to establish an explicit formula for χτ [A;n] with
τ =
[
N − b, 1b
]
. Since the order of the factors of Gn in the character calculation does not
matter (by the conjugate invariance of characters), it will suffice to take A = {1, 2, . . . , ℓ},
for 2 ≤ ℓ ≤ p. In the following, ei denotes the elementary symmetric polynomial of degree i and
the Pochhammer symbol is (a)n =
∏n
i=1(a+i−1). The combined main results are the following.
Theorem 1.1. Let m := p− b− 1, ℓ ≤ p and A ={1, 2, . . . , ℓ}, then
χτ [A;n] =
1∏ℓ
i=1 ni
(1.2)
×
{
min(m,ℓ)∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−k(n1 − 1, n2 − 1, . . . , nℓ − 1) + (−1)ℓ+1 (b− ℓ+ 1)m
m!
}
=
(
b+m
b
)
+
min(b,ℓ−1)∑
i=1
(−1)i
(
b+m− i
b− i
)
ei
(
1
n1
,
1
n2
, . . . ,
1
nℓ
)
. (1.3)
These results come from Theorems 5.9, 5.19 and Proposition 6.1. There are no nonzero
Gn-invariants if m < 0 (as will be seen).
In Section 2, we present general background on spherical functions, harmonic analysis, and
subgroup invariants for finite groups. Section 3 concerns alternating polynomials, which span
a module of isotype
[
N −b, 1b
]
. There is the definition of sums of alternating polynomials which
make up a basis for Gn-invariants. The main results, proving formula (1.2) for the case p = b+1
are in Section 4, and for the cases p ≥ b + 2 are in Section 5, with subsections for p = b + 2
and p > b + 2. The details are of increasing technicality. Section 6 deduces formula (1.3)
from (1.2). Some specializations of the formulas are discussed.
2 Spherical functions
Suppose the representation τ of SN is realized on a linear space V furnished with an SN -invariant
inner product, then there is an orthonormal basis for V in which the restriction toGn decomposes
as a direct sum of irreducible representations of Gn. Suppose the multiplicity of 1Gn is µ, and
the basis is chosen so that for h ∈ Gn
τ(h) =
Iµ O . . . O
O T (1)(h) . . . O
. . . . . . . . . . . .
O O . . . T (r)(h)
,
where T (1), . . . , T (r) are irreducible representations of Gn not equivalent to 1Gn (see [1, Sec-
tions 3.6 and 10] for details). For g ∈ SN , denote the matrix of τ(g) with respect to the basis
4 C.F. Dunkl
by τi,j(g); since τ(h1gh2) = τ(h1)τ(g)τ(h2), we find
1
(#Gn)2
∑
h1,h2∈Gn
τi,j(h1gh2) =
{
τi,j(g), 1 ≤ i, j ≤ µ,
0, else.
Then {τi,j | 1 ≤ i, j ≤ µ} is a basis for the Gn-bi-invariant elements of span{τkℓ}. The spherical
function for the isotype τ and subgroup Gn is defined by
Φτ (g) :=
µ∑
i=1
τii(g), g ∈ SN
(
our notation for χτ [A;n]
)
. Sometimes the term “spherical function” is reserved for Gelfand
pairs where the multiplicity µ = 1. The character of τ is χτ (g) = tr(τ(g)), then
Φτ (g) =
1
#Gn
∑
h∈Gn
χτ (hg).
The symmetrization operator acting on V is (ζ ∈ V )
ρζ :=
1
#Gn
∑
h∈Gn
τ(h)ζ.
The operator ρ is a self-adjoint projection.
Suppose there is an orthogonal subbasis {ψi | 1 ≤ i ≤ µ} for V which satisfies τ(h)ψi = ψi for
h ∈ Gn (thus ρψi = ψi) and 1 ≤ i ≤ µ, then the matrix element τi,i(g) = ⟨τ(g)ψi, ψi⟩/⟨ψi, ψi⟩
and the spherical function Φτ (g) =
∑µ
i=1
1
⟨ψi,ψi⟩⟨τ(g)ψi, ψi⟩. We will produce a formula for Φτ (g)
which works with a non-orthogonal basis of Gn-invariant vectors {ξi | 1 ≤ i ≤ µ} in V. The
Gram matrix M is given by Mij := ⟨ξi, ξj⟩. For g ∈ SN , let T (g)ij := ⟨τ(g)ξj , ξi⟩.
Lemma 2.1. Φτ (g) = tr
(
T (g)M−1
)
.
Proof. Suppose {ζi | 1 ≤ i ≤ µ} is an orthonormal basis for the Gn-invariant vectors in V ,
then there is a (change of basis) matrix [Aij ] such that ζi =
∑µ
j=1Ajiξj and
⟨τ(g)ζi, ζi⟩ =
〈
µ∑
j=1
τ(g)Ajiξj ,
µ∑
k=1
Akiξk
〉
=
∑
j,k
AjiAkiT (g)kj ,
Φτ (g) =
∑
i
∑
j,k
AjiAkiT (g)kj =
∑
j,k
(AA∗)jkT (g)kj = tr((AA∗)T (g)).
Furthermore,
δij = ⟨ζi, ζj⟩ =
∑
k,r
AkiArj⟨ξk, ξr⟩ =
∑
k,r
AkiArjMkr = (A∗MA)ij
and A∗MA = I, M = (A∗)−1A−1 = (AA∗)−1. ■
Corollary 2.2. Suppose ρτ(g)ξi =
∑µ
j=1Bji(g)ξj, 1 ≤ i, j ≤ µ, then Φτ (g) = tr(B(g)).
Proof. The expansion holds because {ξi | 1 ≤ i ≤ µ} is a basis for the invariants. Then
T (g)ij = ⟨τ(g)ξj , ξi⟩ = ⟨ρτ(g)ξj , ξi⟩ =
〈
µ∑
k=1
Bkj(g)ξk, ξi
〉
=
µ∑
k=1
Bkj(g)Mki =
(
MTB(g)
)
ij
and tr
(
T (g)M−1
)
= tr
(
MTB(g)M−1
)
= tr(B(g))
(
note MT =M
)
. ■
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 5
We will use the method provided by the Corollary to determine Φτ (g). This formula avoids
computing T (g) and the inverse M−1 of the Gram matrix. When the multiplicity µ = 1, the
formula simplifies considerably: there is one invariant ψ1, T (g) = ⟨gψ1, ψ1⟩ and M = [⟨ψ1, ψ1⟩]
so that Φτ (g) = ⟨gψ1,ψ1⟩
⟨ψ1,ψ1⟩ (or = c if ρgψ1 = cψ1). We are concerned with computing the spherical
function at a cycle g of length ℓ with no more than one entry from each interval Ij , where the
factor Snj acts only on Ij . The idea is to specify the Gn-invariant polynomials ξ, the effect of
an ℓ-cycle on each of these, and then compute the expansion of ρgξ in the invariant basis.
3 Coordinate systems and invariant sums
of alternating polynomials
To clearly display the action of Gn, we introduce a modified coordinate system. Replace
(x1, x2, . . . , xN ) =
(
x
(1)
1 , . . . , x(1)n1
, x
(2)
1 , . . . , x(2)n2
, . . . , x
(p)
1 , . . . , x(p)np
)
,
that is, x
(j)
i stands for xs with s =
∑j−1
i=1 ni + i. We use x
(i)
∗ , x
(i)
> to denote a generic x
(i)
j with
1 ≤ j ≤ ni, respectively, 2 ≤ j ≤ ni. In the sequel, g denotes the cycle
(
x
(1)
1 , x
(2)
1 , . . . , x
(ℓ)
1
)
.
Throughout, denote m := p− b− 1.
Notation 3.1. For 0 ≤ j ≤ ℓ, denote the elementary symmetric polynomial of degree j in the
variables n1 − 1, n2 − 1, . . . , nℓ − 1 by ej(n∗ − 1). Set πℓ :=
∏ℓ
i=1 ni and πp :=
∏p
i=1 ni. For
integers i ≤ j, the interval {i, i+ 1, . . . , j} ⊂ N is denoted by [i, j].
Definition 3.2. The action of the symmetric group SN on polynomials P (x) is given by
wP (x) = P (xw) and (xw)i = xw(i), w ∈ SN , 1 ≤ i ≤ N .
Note (x(vw))i = (xv)w(i) = xv(w(i)) = xvw(i), vwP (x) = (wP )(xv) = P (xvw). The projection
onto Gn-invariant polynomials is given by
ρP (x) =
1
#Gn
∑
h∈Gn
P (xh).
We use the polynomial module of isotype
[
N − b, 1b
]
with the lowest degree. This module is
spanned by alternating polynomials in b+ 1 variables.
Definition 3.3. For xi1 , xi2 , . . . xib+1
, let
∆
(
xi1 , xi2 , . . . xib+1
)
:=
∏
1≤j<k≤b+1
(
xij − xik
)
.
Lemma 3.4. Suppose x1, . . . , xb+2 are arbitrary variables and fj := ∆(x1, x2,, . . . , x̂j , . . . , xb+2)
denotes the alternating polynomial when xj is removed from the list, then
∑b+2
j=1(−1)jfj = 0.
Proof. Let F (x) :=
∑b+2
j=1(−1)jfj . Suppose 1 ≤ i < b + 2 and (i, i + 1) is the trans-
position of xi and xi+1, then (i, i + 1)fj = −fj if j ̸= i, i + 1, (i, i + 1)fi = fi+1 and
(i, i+1)fi+1 = fi. Thus (i, i+ 1)F (x) = −F (x) and this implies F (x) is divisible by the alternat-
ing polynomial in x1, . . . , xb+2 which is of degree 1
2(b+ 2)(b+ 1), but F is of degree ≤ 1
2b(b+ 1)
and hence F (x) = 0. ■
We briefly discuss the relation between hook tableaux and alternating polynomials. The irre-
ducible representation of SN corresponding to the partition
[
N − b, 1b
]
has the basis of standard
Young tableaux of this shape. For a given tableau T and an entry i (with 1 ≤ i ≤ N) the content
6 C.F. Dunkl
c(i, T ) := col(i, T ) − row(i, T ) (the labels of the column, row of T containing i). The Jucys–
Murphy elements ωj :=
∑j−1
i=1 (i, j), 1 ≤ j ≤ N , mutually commute and satisfy ωjT = c(j, T )T ;
this is the defining property of the representation. There is a general identity ωj+1 = σjωjσj+σj
where σj := (j, j+1) for 1 ≤ j < N . The tableau T0 with first column containing {1, 2, . . . , b+1}
satisfies c(i, T0) = 1− i for 1 ≤ i ≤ b+1 and = i− b− 1 for b+2 ≤ i ≤ N . We use the notation
of Lemma 3.4 so that fb+2 := ∆(x1, . . . , xb+1).
Proposition 3.5. ωifb+2 = c(i, T0)fb+2 for 1 ≤ i ≤ N .
Proof. Since (i, j)fb+2 = −fb+2 for 1 ≤ i < j ≤ b+ 1, it follows that ωifb+2 = −(i− 1)fb+2 for
1 < i ≤ b+ 1, while ω1 = 0 implies ω1fb+2 = 0 = c(1, T0)fb+2. The claim ωb+2fb+2 = fb+2 needs
more detail. Consider the term for (i, b+ 2) in ωb+2fb+2 (for 1 ≤ i ≤ b+ 1)
(i, b+ 2)∆(x1, . . . xb+1) = ∆
(
x1, . . . ,
(i)
xb+2, . . . , xb+1
)
= (−1)b+1−ifi,
the sign comes from moving xb+2 by b + 1 − i adjacent transpositions to the last argument
of ∆(∗). Thus
b+1∑
i=1
(i, b+ 2)fb+2 =
b+1∑
i=1
(−1)b+1−ifi = fb+2
by the lemma
(
multiply each term by (−1)b+1
)
. Now let i ≥ b + 2 and suppose ωifb+2 =
(i − b − 1)fb+2, then ωi+1fb+2 = (σiωiσi + σi)fb+2 = (ωi + 1)fb+2 = (i − b)fb+2. By induction,
ωifb+2 = (i− b− 1)fb+2 for b+ 2 ≤ i ≤ N and this completes the proof. ■
The SN -module spanned by fb+2 is of isotype
[
N − b, 1b
]
.
Usually x denotes a (b+ 1)-tuple as a generic argument of ∆.
Definition 3.6. Suppose x =
(
x
(j1)
i1
, x
(j2)
i2
, . . . , x
(jb+1)
ib+1
)
, then
L(x) := (j1, j2, . . . , jb+1) and ∆(x) := ∆
(
x
(j1)
i1
, x
(j2)
i2
, . . . , x
(jb+1)
ib+1
)
.
The arguments in L(x) can be assumed to be in increasing order, up to a change in sign
of ∆(x) (for example, if σ is a transposition, then ∆(xσ) = −∆(x)).
Proposition 3.7. If h ∈ Gn, then L(xh) = L(x), and
ρ∆(x) =
b+1∏
r=1
n−1
jr
∑
{∆(y) | L(y) = L(x)}.
Proof. The second statement follows from the multiplicative property of ρ and from
1
nj !
∑
h∈Snj
f
(
x
(j)
∗ h
)
=
1
nj
nj∑
i=1
f
(
x
(j)
i
)
,
where x
(j)
∗ denotes an arbitrary x
(j)
i , and (nj − 1)! elements h fix x
(j)
i . ■
It follows from Lemma 3.4 that a basis for the Gn-invariant polynomials is generated from
∆(x) with x =
(
x
(j1)
i1
, x
(j2)
i2
, . . . , x
(p)
ib+1
)
, that is, the last coordinate is in Ip.
In the following, we specify invariants by the indices omitted from L(x); this is actually more
convenient.
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 7
Definition 3.8. Suppose S ⊂ [1, p− 1] with #S = m, then define the invariant polynomial
ξS :=
∑{
∆(x) | x =
(
. . . , x
(j)
ij
, . . . , x
(p)
ip
)
, j /∈ S
}
.
That is, the coordinates of x have b distinct indices from [1, b + m] \ S. The basis has
cardinality µ =
(
b+m
b
)
(see [5, p. 105, Example 2 (b)]). The underlying task is to compute the
coefficient BS,S(g) in ρgξS =
∑
S′ BS′,S(g)ξS′ . This requires a decomposition of ξS .
Definition 3.9. Suppose S ⊂ [1, p − 1] with #S = m and E ⊂ ([1, ℓ] ∪ {p}) \ S say x ∈ XS,E
if j ∈ E implies xj = x
(j)
1 , j ∈ [ℓ+1, p]\S implies xj = x
(j)
k with 1 ≤ k ≤ nj and j ∈ [1, ℓ]\(S∪E)
implies xj = x
(j)
k with 2 ≤ k ≤ nj (consider x as a (b+1)-tuple indexed by [1, b+m] \S ∪ {p}).
Furthermore, let
ϕS,E :=
∑
{∆(x) | x ∈ XS,E}.
Thus ξS =
∑
E ϕS,E and we will analyze ρgϕS,E . It turns out only a small number of
sets E allow ρgϕS,E ̸= 0, and an even smaller number have a nonzero coefficient BS,(S,E) in the
expansion ρgϕS,E =
∑
S′ BS′,(S,E)ξS′ , namely ∅ (the empty set), [1, ℓ] and [1,minS − 1] ∪ {p}.
Part of the discussion is to show this list is exhaustive.
Lemma 3.10. For ρ∆(x) ̸= 0, it is necessary that there be no repetitions in L(x).
Proof. Suppose ja = jb = k, and ∆
(
. . . , x
(k)
ia
, . . . , x
(k)
ib
, . . .
)
appears in the sum; we can as-
sume b = a + 1 by rearranging the variables, possibly introducing a sign factor. If ia =
ib, then ∆(x) = 0, else ∆
(
. . . , x
(k)
ib
, x
(k)
ia
, . . .
)
also appears, by the action of the transposition
(ia, ib) ∈ Snk
These two terms cancel out because ∆(x(ia, ib)) = −∆(x). ■
If some x has coordinates x
(i)
1 , x
(i+1)
> and i < ℓ, then ρg∆(x) = 0 by the lemma since
xg =
(
. . . , x
(i+1)
1 , x
(i+1)
> , . . .
)
. This strongly limits the sets E allowing ρ∆(xg) ̸= 0.
4 The case p = b + 1
This is the least complicated situation and introduces some techniques used later. Here L(x) =
(1, 2, . . . , b, b+ 1). There is just one Gn-invariant polynomial (up to scalar multiplication):
ψ :=
∑{
∆
(
x
(1)
i1
, x
(2)
i2
, . . . , x
(b+1)
ip
)
| 1 ≤ i1 ≤ n1, 1 ≤ i2 ≤ n2, . . . , 1 ≤ ib+1 ≤ nb+1
}
.
In Definition 3.9, the set S = ∅ and we write ϕE for ϕ∅,E .
Proposition 4.1. If 1 ≤ #E < ℓ, then ρgϕE = 0.
Proof. Suppose there are indices k, k + 1 with k ∈ E, k < ℓ and k + 1 /∈ E. If ∆(x) is one of
the summands of ϕE , then x =
(
. . . , x
(k)
1 , x
(k+1)
> , . . .
)
and xg =
(
. . . , x
(k+1)
1 , x
(k+1)
> , . . .
)
and by
Lemma 3.10 ρ∆(x) = 0. Otherwise k ∈ E implies k + 1 ∈ E or k = ℓ which by hypothesis
implies ℓ ∈ E and 1 /∈ E, then x =
(
x
(1)
> . . . , x
(ℓ)
1 , . . .
)
, xg =
(
x
(1)
> . . . , x
(1)
1 , . . .
)
and ρ∆(x) = 0
as before. ■
It remains to compute ρgϕE for E = ∅ and E = [1, ℓ]. Note gϕ∅ = ϕ∅.
Suppose x ∈ X∅,∅, then x =
(
x
(1)
> , . . . , x
(ℓ)
> , x
(ℓ+1)
∗ , . . . , x
(b+1)
∗
)
, and since ρ∆(xg) = ρ∆(x) =
1
πp
ψ (by Proposition 3.7) and #X∅,∅ =
∏ℓ
i=1(ni − 1)
∏b+1
j=ℓ+1 nj , it follows that
ρϕ∅,∅ =
1
πℓ
ℓ∏
i=1
(ni − 1) =
1
πℓ
eℓ(n∗ − 1).
8 C.F. Dunkl
Now suppose E = [1, ℓ] and x ∈ X∅,[1,ℓ] implies x =
(
x
(1)
1 , . . . , x
(ℓ)
1 , x
(ℓ+1)
∗ , . . . , x
(b+1)
∗
)
, then
xg =
(
x
(ℓ)
1 , x
(1)
1 , . . . , x
(ℓ−1)
1 , x
(ℓ+1)
∗ , . . . , x
(b+1)
∗
)
.
Applying ℓ − 1 transpositions (ℓ − 1, ℓ), (ℓ − 2, ℓ − 1), . . . , (1, 2) transforms x to xg and thus
∆(xg) = (−1)ℓ−1∆(x). So
ρ∆(xg) = (−1)ℓ−1ρ∆(x) = (−1)ℓ−1 1
πp
ψ.
Since #X∅,[1,ℓ] =
∏b+1
i=ℓ+1 ni, it follows that ρϕB,[1,ℓ] =
(−1)ℓ−1
πℓ
.
Proposition 4.2. Suppose p = b+ 1 and 2 ≤ ℓ ≤ b+ 1, then
Φτ (g) =
1
πℓ
{
eℓ(n∗ − 1) + (−1)ℓ−1
}
.
Proof. ρgψ = ρgϕ∅,∅ + ρgϕ∅,[1,ℓ] =
1
πℓ
{
eℓ(n∗ − 1) + (−1)ℓ−1
}
ψ. ■
This is the main formula (1.2) specialized to m = 0.
5 The cases p > b + 1
There is some simplification for p = b+ 2 compared to p ≥ b+ 3. First, we set up some tools.
Definition 5.1. For an invariant basis element ξ and a polynomial ϕ, let coef(ξ, ρϕ) denote the
coefficient of ξ in the expansion of ρϕ in the basis.
The main object is to determine
Φτ (g) =
∑
S
coef(ξS , ρgξS). (5.1)
Proposition 5.2. Suppose S and E are given by Definition 3.9, ℓ ≤ b+m, coef(ξS , ρgϕS,E) ̸= 0,
then E = ∅ or S ∩ [1, ℓ] = ∅ and E = [1, ℓ].
Proof. Let υE := {j ∈ E | j + 1 /∈ E}, the upper end-points of E. If j ∈ υE and j + 1 ∈
[1, b+m] \ (S ∪ E), then L(xg) = (. . . , j + 1, j + 1, . . . ) and ρg∆(x) = 0 (Proposition 3.10)
Thus x ∈ XS,E , ρg∆(x) ̸= 0 and k ∈ υE implies k + 1 ∈ S ∪ {ℓ}. Suppose j ∈ υE, j + 1 ∈ S
and j < ℓ, then x
(j+1)
1 appears in gϕE . That is, if x ∈ XS,E , then j + 1 is not an entry of L(x)
but j + 1 appears in L(xg)) and thus coef(ξS , ρgϕS,E) = 0. Another possibility is that there
exists i /∈ S ∪ E, 1 ≤ i < ℓ and i + 1 ∈ E, in which case i + 1 does not appear in L(xg)
and coef(ξS , ρgϕS,E) = 0. ■
The case ℓ = b+m+ 1 involves more technicalities.
Informally, consider S as the set of holes in L(x); no new holes can be adjoined or removed
from L(xg) because this would imply coef(ξS , ρ∆(xg)) = 0. And of course L(xg) can have
no repetitions. This is the idea that limits the possible boundary points of E (that is, j ∈ E
and j + 1 /∈ E or j − 1 /∈ E).
Recall ϕS,E :=
∑
{∆(x) | x ∈ XS,E}, and the task is to determine coef(ξS , ρgϕS,E).
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 9
5.1 Case p = b + 2
Here m = 1 so the sets S are singletons {i} with 1 ≤ i ≤ b+1. Note that m = 1 is an underlying
hypothesis throughout this subsection. Write ξi in place of ξ{i}. Then {ξi | 1 ≤ i ≤ b + 1}
is a basis for the Gn-invariants. The possibilities for E are ∅ for any i, [1, ℓ] for i > ℓ, and
[1, i− 1] ∪ {b+ 2} for ℓ = b+ 2. Suppose E = ∅, then x ∈ Xi,∅ implies
x =
(
x
(1)
> , . . . , x
(ℓ)
> , x
(ℓ+1)
∗ , . . . , x
(b+2)
∗
)
with x
(i)
> , x
(i)
∗ omitted if i ≤ ℓ or i > ℓ, respectively. From Proposition 3.7,
ρ∆(xg) = ρ∆(x) =
ni
πp
ξi.
Also #Xi,∅ =
∏ℓ
j=1,j ̸=i(nj − 1)
∏b+2
k=ℓ+1 nk, or
∏ℓ
j=1(nj − 1)
∏b+2
k=ℓ+1,k ̸=i nk, if i ≤ ℓ or i > ℓ,
respectively.
Proposition 5.3. Suppose 2 ≤ ℓ ≤ b+ 1 and i > ℓ, then coef(ξi, ρϕi,∅) =
1
πℓ
eℓ(n∗ − 1).
Proof. Multiply #Xi,∅ by ni
πp
with result 1
πℓ
∏ℓ
j=1(nj − 1). ■
Proposition 5.4. Suppose 2 ≤ ℓ ≤ b+2 and i ≤ ℓ, then coef(ξi, ρgϕi,∅) =
1
πℓ
eℓ(n∗−1)
(
1+ 1
ni−1
)
.
Proof. Multiply #Xi,∅ by ni
πp
with result
ℓ∏
j=1,j ̸=i
nj − 1
nj
=
(
ℓ∏
j=1
nj − 1
nj
)(
ni
ni − 1
)
=
1
πℓ
ℓ∏
j=1
(nj − 1)
(
1 +
1
ni − 1
)
. ■
Proposition 5.5. Suppose 2 ≤ ℓ ≤ b+ 2 and ℓ < i, then coef
(
ξi, ρgϕi,[1,ℓ]
)
= 1
πℓ
(−1)ℓ−1.
Proof. If x ∈ Xi,[1,ℓ], then x =
(
x
(1)
1 , . . . , x
(ℓ)
1 , x
(ℓ+1)
∗ , . . . , x
(b+2)
∗
)
omitting x
(i)
∗ and L(xg) =
(2, 3, . . . , ℓ, 1, ℓ+ 1, . . . , i− 1, i+ 1, . . . , b+ 2). Applying a product of ℓ− 1 transpositions shows
that ∆(xg) = (−1)ℓ−1∆(x) and ρ∆(xg) = (−1)ℓ−1 ni
πp
ξi. Multiply (−1)ℓ−1 ni
πp
by #Xi,[1,ℓ] =∏b+2
s=ℓ+1,s ̸=i ns to obtain (−1)ℓ−1
∏ℓ
s=1 n
−1
s . ■
Theorem 5.6. Suppose 2 ≤ ℓ ≤ b+ 1, then
Φτ (g) =
1
πℓ
{
(b+ 1)eℓ(n∗ − 1) + eℓ−1(n∗ − 1) + (−1)ℓ−1(b− ℓ+ 1)
}
.
Proof. Break up the sum (5.1) into i > ℓ and i ≤ ℓ sums:
b+1∑
i=ℓ+1
coef(ξi, ρgξi) =
b+1∑
i=ℓ+1
(
coef(ξi, ρgϕi,∅) + coef
(
ξi, ρgϕ[1,ℓ]
))
=
1
πℓ
(b+ 1− ℓ)
(
eℓ(n∗ − 1) + (−1)ℓ−1
)
,
ℓ∑
i=1
coef(ξi, ρgξi) =
ℓ∑
i=1
coef(ξi, ρgϕi,∅) =
1
πℓ
ℓ∑
i=1
eℓ(n∗ − 1)
(
1 +
1
ni − 1
)
=
1
πℓ
(ℓeℓ(n∗ − 1) + eℓ−1(n∗ − 1)).
Add the two parts together. ■
10 C.F. Dunkl
Proposition 5.7. Suppose ℓ = b+ 2 and 1 ≤ i ≤ b+ 1, then for E = [1, i− 1] ∪ {b+ 2}
coef(ξi, ρgϕi,E) =
1
πp
(−1)i−1
b+1∏
s=i+1
(ns − 1).
Proof. If x ∈ Xi,E , then xg =
(
x
(2)
1 , . . . , x
(i)
1 , x
(i+1)
> , . . . , x
(1)
1
)
, L(xg) = (2, . . . , i, i + 1, . . . ,
b+ 1, 1) and ∆(xg) = (−1)b∆(y) with L(y) = (1, 2, . . . , b+ 1). Apply Lemma 3.4 to obtain
b∑
j=1
(−1)j∆(x1, x2,, . . . , x̂j , . . . , xb+2) + (−1)b+2∆(x1, x2,, . . . , xb+1) = 0
(the notation x̂j means xj is omitted). Use the term j = i in the identity to obtain
coef(ξi, ρg∆(y)) =
ni
πp
(−1)b+1−i.
From #Xi,E =
∏b+1
s=i+1(ns − 1), it follows that
coef(ξi, ρgϕi,E) =
1
πp
(−1)i−1
b+1∏
s=i+1
(ns − 1). ■
Proposition 5.8. Suppose ℓ = b+ 2, then
b+1∑
i=1
coef
(
ξi, ρgϕi,[1,i−1]∪{p}
)
=
1
πp
{
b+1∏
s=1
(ns − 1)− (−1)b
}
.
Proof. The sum is
1
πp
b+1∑
i=1
(−1)i−1ni
b+1∏
s=i+1
(ns − 1) =
1
πp
b+1∑
i=1
(−1)i−1(ni − 1 + 1)
b+1∏
s=i+1
(ns − 1)
=
1
πp
b+1∑
i=1
{
(−1)i−1
b+1∏
s=i
(ns − 1)− (−1)i
b+1∏
s=i+1
(ns − 1)
}
=
1
πp
b+1∏
s=1
(ns − 1)− 1
πp
(−1)b+1
by telescoping, leaving the first product with i = 1 and the last with i = b+ 1. ■
Theorem 5.9. Suppose ℓ = b+ 2, then
Φτ (g) =
1
πℓ
{
(ℓ− 1)eℓ(n∗ − 1) + eℓ−1(n∗ − 1) + (−1)b
}
.
Proof. Combine Propositions 5.4 and 5.8 (note πp = πℓ),
b+1∑
i=1
coef(ξi, ρgξi) =
1
πℓ
ℓ−1∑
i=1
ℓ∏
j=1
(nj − 1)
(
1 +
1
ni − 1
)
+
1
πp
ℓ−1∏
s=1
(ns − 1)− 1
πp
(−1)b+1
=
1
πℓ
ℓ∏
j=1
(nj − 1)
{
ℓ−1∑
i=1
(
1 +
1
ni − 1
)
+
1
nℓ − 1
}
− 1
πp
(−1)b+1
=
1
πℓ
{
(ℓ− 1)eℓ(n∗ − 1) + eℓ−1(n∗ − 1) + (−1)b
}
. ■
Formula (1.2) with m = 1, ℓ = b+ 2 has the term (−1)ℓ+1 (b−ℓ+1)m
m! = (−1)ℓ+2 = (−1)b. This
completes the case p = b+ 2.
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 11
5.2 Case p > b + 2
Write p = b+m+ 1. Label the invariants by S ⊂ [1, b+m], #S = m,
ξS :=
∑{
∆
(
x
(j1)
i1
, x
(j2)
i2
, . . . , x
(jb)
ib
, x
(p)
ip
)
| {j1, . . . , jb} = [1, b+m] \ S,
1 ≤ is ≤ njs , 1 ≤ s ≤ b, 1 ≤ ip ≤ np
}
,
and in ∆(x) take j1 < j2 < · · · < jb. The following lemma generalizes the generating function
for elementary symmetric polynomials.
Lemma 5.10. Suppose y1, y2, . . . , yr are variables and q ≤ r ≤ s, then
r∏
i=1
yi
∑
U⊂[1,s],#U=q
∏
j∈U∩[1,r]
(
1 +
1
yj
)
=
min(q,r)∑
k=0
(
s− k
q − k
)
er−k(y1, . . . , yr).
Proof. The product∏
j∈U∩[1,r]
(
1 +
1
yj
)
=
q∑
k=0
∑{∏
j∈V
(
1
yj
)
| V ⊂ U ∩ [1, r], #V = k
}
.
Any particular V with #V = k appears in
(
s−k
q−k
)
different sets U . Then ek
(
y−1
1 , . . . , y−1
r
)
is
a sum of
∏
j∈V
(
1
yj
)
over k-subsets of [1, r], and thus the sum is
∑min(q,r)
k=0
(
s−k
q−k
)
ek
(
y−1
1 , . . . , y−1
r
)
.
Also (
r∏
i=1
yi
)
ek
(
y−1
1 , . . . , y−1
r
)
= er−k(y1, . . . , yr). ■
The apparent singularity at yj = 0 is removable.
Proposition 5.11. If S ⊂ [1, b+m], #S = m and ℓ ≤ b+m+ 1, then
coef(ξS , ρgϕS,∅) =
∏{
ni − 1
ni
| 1 ≤ i ≤ ℓ, i /∈ S
}
.
Proof. When E = ∅, then x ∈ XS,E satisfies ρ∆(xg) = ρ∆(x) =
(∏b+m+1
j=1,j /∈Sn
−1
j
)
ξS . Further-
more, #XS,∅ =
∏ℓ
i=1,i/∈S(ni − 1)×
∏b+m+1
j=ℓ+1,j /∈Snj and the product of the two factors is
ℓ∏
i=1,i/∈S
(
ni − 1
ni
)
. ■
Proposition 5.12. For ℓ ≤ b+m,
∑
S⊂[1,b+m],#S=m
coef(ξS , ρgϕS,∅) =
1
πℓ
min(m,ℓ)∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−k(n∗ − 1).
Proof. The sum equals∑
S⊂[1,b+m],#S=m
ℓ∏
s=1,i/∈S
(
ns − 1
ns
)
=
1
πℓ
ℓ∏
i=1
(ni − 1)
∑
S⊂[1,b+m],#S=m
ℓ∏
j∈[1,ℓ]∩S
(
nj
nj − 1
)
=
1
πℓ
min(m,ℓ)∑
k=0
(
b+m− k
m− k
)
eℓ−k(n∗ − 1)
by Lemma 5.10 with r = ℓ, s = b+m, q = m and yi = ni − 1. Also(
b+m− k
m− k
)
=
(b+m− k)!
b!(m− k)!
=
(b+ 1)m−k
(m− k)!
. ■
12 C.F. Dunkl
Proposition 5.13. If ℓ ≤ b, S ⊂ [ℓ+ 1, b+m], #S = m, then coef
(
ξS , ρgϕS,[1,ℓ]
)
= (−1)ℓ+1
πℓ
.
Proof. If x ∈ XS,[1,ℓ], then L(xg) = (2, 3, . . . , ℓ, 1, ℓ+1, . . . , b+m+1) with {j | j ∈ S} omitted.
Thus
∆(xg) = (−1)ℓ−1∆(x) and ρ∆(xg) = (−1)ℓ−1
b+m+1∏
i=1,i/∈S
n−1
i ξS .
The number of summands in ϕS,[1,ℓ] is
#XS,[1,ℓ] =
b+m+1∏
s=ℓ+1,s/∈S
ns
and the required coefficient is the product with (−1)ℓ−1
∏b+m+2
i=1,i/∈Sn
−1
i , namely
(−1)ℓ−1
ℓ∏
i=1
n−1
i . ■
Theorem 5.14. Suppose ℓ ≤ b+m, then
Φτ (g) =
1
πℓ
{
min(m,ℓ)∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−k(n∗ − 1) + (−1)ℓ+1 (b− ℓ+ 1)m
m!
}
.
Proof. When b < ℓ ≤ b+m, then the sum (5.1) equals
∑
S coef(ξS , ρgϕS,∅), else if 2 ≤ ℓ ≤ b,
then it equals∑
S⊂[1,b+m]
coef(ξS , ρgϕS,∅) +
∑
S⊂[ℓ+1,b+m]
coef(ξS , ρgϕS,[1,ℓ]).
There are
(
b+m−ℓ
m
)
subsets S ⊂ [ℓ+1, b+m]. In both cases the sums evaluate to the claimed value,
since (b−ℓ+1)m
m! =
(
b+m−ℓ
m
)
if ℓ ≤ b and = 0 if b+ 1 ≤ ℓ ≤ b+m. ■
For the case ℓ = b+m+1, the sets E which allow coef(ξS , ρgϕS,E) ̸= 0 are E = ∅, [1,minS−
1] ∪ {ℓ}. Lemma 3.4 is used just as in the situation m = 1.
Proposition 5.15. Suppose ℓ = b+m+ 1, then∑
S⊂[1,b+m],#S=m
coef(ξS , ρgϕS,∅) =
1
πℓ
(nℓ − 1)
m∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−1−k(n1 − 1, . . . , nℓ−1 − 1).
Proof. By Proposition 5.11,
coef(ξS , ρgϕS,∅) =
b+m+1∏
i=1,i/∈S
ni − 1
ni
=
(
b+m+1∏
i=1
ni − 1
ni
)∏
j∈S
(
nj
nj − 1
)
and ∑
S⊂[1,b+m],#S=m
coef(ξS , ρgϕS,∅) =
1
πℓ
ℓ∏
i=1
(ni − 1)
∑
S⊂[1,b+m]
∏
j∈S
(
1 +
1
nj − 1
)
= (nℓ − 1)
1
πℓ
b+m∏
i=1
(ni − 1)
∑
S⊂[1,b+m]
∏
j∈S
(
1 +
1
nj − 1
)
=
1
πℓ
(nℓ − 1)
m∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−1−k(n1 − 1, . . . , nℓ−1 − 1),
by Lemma 5.10 with r = s = b+m, q = m, (r = ℓ− 1) and yi = ni − 1. ■
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 13
Proposition 5.16. Suppose S ⊂ [1, b+m], #S = m and ℓ = b+m+1, and E = [1, t− 1]∪{ℓ}
with t := minS, then
coef(ξS , ρgϕS,E) = (−1)t+1 1
πℓ
b+m∏
i=t
(ni − 1)
∏
j∈S
(
1 +
1
nj − 1
)
.
Proof. If minS = 1, then E = {ℓ}. Set t := minS. A typical point in XS,E is
x =
(
x
(1)
1 , x
(2)
1 , . . . , x
(t−1)
1 , x
(t+1)
> , . . . , x
(ℓ−1)
> , x
(ℓ)
1
)
omitting
{
x
(j)
∗ | j ∈ S
}
. Then
xg =
(
x
(2)
1 , x
(3)
1 , . . . , x
(t)
1 , x
(t+1)
> , . . . , x
(ℓ−1)
> , x
(1)
1
)
and
∆(xg) = (−1)b∆
(
x
(1)
1 , x
(2)
1 , . . . , x
(t)
1 , x
(t+1)
> , . . . , x
(ℓ−1)
>
)
omitting S \ {t} terms (applying b adjacent transpositions). To apply Lemma 3.4, we relabel(
x
(1)
1 , . . . , x
(t)
1 , x
(t+1)
> , . . . , x
(ℓ−1)
> , x
(ℓ)
1
)
(omit S \ {t}) as (y1, . . . , yb+2) with yi = x
(i)
1 for 1 ≤ i ≤ t
and i = ℓ. Thus
∆(y1, y2,, . . . , yb+1) =
b∑
j=1
(−1)j+b+1∆(y1, y2,, . . . , ŷj , . . . , yb+2).
Apply ρ, then the term with j = t becomes (−1)t+b+1
(∏ℓ
i=1,i/∈S n
−1
i
)
ξS . Thus
coef(ξS , ρ∆(xg)) = (−1)b(−1)t+b+1
(
ℓ∏
i=1,i/∈S
n−1
i
)
.
Multiply by #XS,E =
∏b+m
i=t+1,i/∈S(ni − 1) to obtain
coef(ξS , ρgϕS,E) = (−1)t+1 1
πℓ
b+m∏
i=t
(ni − 1)
∏
j∈S
(
1 +
1
nj − 1
)
. ■
The next step is to sum over S with the same minS.
Proposition 5.17. Suppose ℓ = b+m+ 1, 1 ≤ t ≤ b+ 1, and E = [1, t− 1] ∪ {ℓ}, then
∑
minS=t
coef(ξS , ρgϕS,E) = (−1)t
nt
πℓ
ℓ−1−t∑
k=0
(m− k)b+1−t
(b+ 1− t)!
eℓ−1−t−k(nt+1 − 1, . . . , nℓ−1 − 1).
Proof. By Proposition 5.16,
∑
minS=t
coef(ξS , ρgϕS,E) = (−1)t+1 1
πℓ
b+m∏
i=t
(ni − 1)
(
1 +
1
nt − 1
)
×
∑
U⊂[t+1,b+m],#U=m−1
∏
j∈U
(
1 +
1
nj − 1
)
= (−1)t
nt
πℓ
m−1∑
k=0
(
b+m− t− k
m− 1− k
)
eℓ−1−t−k(nt+1 − 1, . . . , nℓ−1 − 1)
by Lemma 5.10 with r = s = ℓ − 1 − t, q = m − 1 (note b + m = ℓ − 1). In the inner sum
S = U ∪ {t}. The binomial coefficient is equal to the coefficient in the claim. ■
14 C.F. Dunkl
To shorten some ensuing expressions, introduce
e(k;u) := ek(nu − 1, nu+1 − 1, . . . , nℓ−1 − 1).
Proposition 5.18. Suppose ℓ = b+m+ 1, then
b+1∑
t=1
(−1)tnt
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
e(ℓ− 1− t− k; t+ 1) =
m∑
k=1
(b+ 1)m−k
(m− k)!
e(ℓ− k; 1) + (−1)b.
Proof. Write nt = (nt − 1) + 1 and use a simple identity for elementary symmetric functions
nte(ℓ− 1− t− k; t+ 1) = e(ℓ− t− k; t)− e(ℓ− t− k; t+ 1) + e(ℓ− 1− t− k; t+ 1),
then the sum becomes
m−1∑
k=0
(m− k)b
b!
e(ℓ− 1− k; 1)
+
b+1∑
t=2
(−1)t+1
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
e(ℓ− t− k; t) (5.2)
−
b+1∑
t=1
(−1)t+1
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
e(ℓ− t− k; t+ 1) (5.3)
+
b+1∑
t=1
(−1)t+1
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
e(ℓ− 1− t− k; t+ 1),
We will show that there is a three-term telescoping effect, after changing the summation variables
in sums: (5.2) t→ t+ 1, (5.3) k → k + 1. This results in (displayed in same order)
m−1∑
k=0
(m− k)b
b!
e(ℓ− 1− k; 1)
+
b+1∑
t=1
(−1)t
m−1∑
k=0
(m− k)b−t
(b− t)!
e(ℓ− 1− t− k; t+ 1)
−
b+1∑
t=1
(−1)t+1
m−2∑
k=−1
(m− k − 1)b+1−t
(b+ 1− t)!
e(ℓ− 1− t− k; t+ 1)
+
b+1∑
t=1
(−1)t+1
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
e(ℓ− 1− t− k; t+ 1).
Thus the coefficient of e(ℓ− 1− t− k; t+ 1) is
b∑
t=1
(−1)t
m−1∑
k=0
(m− k)b−t
(b− t)!
−
b+1∑
t=1
(−1)t+1
m−2∑
k=−1
(m− k − 1)b+1−t
(b+ 1− t)!
+
b+1∑
t=1
(−1)t+1
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
,
the limits in the middle sum can be replaced by 0 ≤ k ≤ m − 2 since e(ℓ − t; t + 1) = 0
(at k = −1). If a pair (t, k) occurs in each sum, then the sum of these terms vanishes, by
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 15
a straightforward calculation. Exceptions are at t = b + 1 (where ℓ − 1 − b − 1 = m − 1
and e(ℓ− 1− t− k; t+ 1) = e(m− 1− k; b+ 2)) and at 1 ≤ t ≤ b, k = m− 1
(−1)b+2
{
−
m−2∑
k=0
1 +
m−1∑
k=0
1
}
e(m− 1− k; b+ 2) = (−1)be(0; b+ 2),{
b∑
t=1
(−1)t +
b∑
t=1
(−1)t+1
}
e(ℓ− t−m; t+ 1) = 0,
respectively. Thus
b+1∑
t=1
(−1)tnt
m−1∑
k=0
(m− k)b+1−t
(b+ 1− t)!
e(ℓ− 1− t− k; t+ 1)
=
m−1∑
k=0
(m− k)b
b!
e(ℓ− 1− k; 1) + (−1)b
=
m∑
k=1
(b+ 1)m−k
(m− k)!
e(ℓ− k; 1) + (−1)b,
(changing k → k − 1) since (m−k)b
b! =
(
m−k−1+b
m−k−1
)
=
(b+1)m−k−1
(m−k−1)! . ■
Theorem 5.19. Suppose ℓ = b+m+ 1(= p), then
Φτ (g) = π−1
ℓ
{
m∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−k(n∗ − 1) + (−1)b
}
.
Proof. Combining the values from Proposition 5.15 for E = ∅ and from Proposition 5.18 for
E = [1,minS − 1] ∪ {p},
∑
S
coef(ξS , ρgξS) =
1
πℓ
(nℓ − 1)
m∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−1−k(n1 − 1, . . . , nℓ−1 − 1)
+
1
πℓ
{
m∑
k=1
(b+ 1)m−k
(m− k)!
eℓ−k(n1 − 1, . . . , nℓ−1 − 1) + (−1)b
}
=
1
πℓ
{
m∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−k(n∗ − 1) + (−1)b
}
.
In the second line the lower limit k = 1 can be replaced by k = 0 because
eℓ(n1 − 1, . . . , nℓ−1 − 1) = 0. ■
Observe that formula (1.2) contains (−1)ℓ+1 (b−ℓ+1)m
m! which becomes
(−1)ℓ+1 (−m)m
m!
= (−1)m+ℓ+1,
and ℓ = b+m+1. Thus we have proven the general formula for any ℓ with 2 ≤ ℓ ≤ p = b+m+1.
16 C.F. Dunkl
6 An equivalent formula
Formula (1.2) can be expressed in terms of ek
(
1
n1
, . . . , 1
nℓ
)
, as displayed in formula (1.3).
Proposition 6.1. For m ≥ 0 and 2 ≤ ℓ ≤ b+m+ 1 = p,
1
πℓ
{
min(m,ℓ)∑
k=0
(b+ 1)m−k
(m− k)!
eℓ−k(n∗ − 1) + (−1)ℓ+1 (b− ℓ+ 1)m
m!
}
=
(
b+m
b
)
+
min(b,ℓ−1)∑
i=1
(−1)i
(
b+m− i
b− i
)
ei
(
1
n1
, . . . ,
1
nℓ
)
.
Proof. From the generating function for elementary symmetric functions (we denote ei(n1, n2,
. . . , nℓ) by ei(n∗)),
ℓ∑
j=0
tjej(n∗ − 1) =
ℓ∏
i=1
(1 + t(ni − 1)) = (1− t)ℓ
ℓ∏
i=1
(
1 +
t
1− t
ni
)
=
ℓ∑
i=1
(1− t)ℓ−itiei(n∗) =
ℓ∑
i=1
ℓ−i∑
k=0
(−1)k
(
ℓ− i
k
)
ti+kei(n∗)
=
ℓ∑
j=0
tj
j∑
i=0
(−1)j−i
(
ℓ− i
j − i
)
ei(n∗),
and thus ej(n∗ − 1) =
j∑
i=0
(−1)j−i
(
ℓ−i
j−i
)
ei(n∗). The first formula equals
π−1
ℓ
{
min(m,ℓ)∑
k=0
ℓ−k∑
i=0
(b+ 1)m−k
(m− k)!
(−1)ℓ−k−i
(
ℓ− i
ℓ− k − i
)
ei(n∗) + (−1)ℓ+1 (b− ℓ+ 1)m
m!
}
;
the coefficient of ei(n∗) is
min(m,ℓ−i)∑
k=0
(b+ 1)m−k
(m− k)!
(i− ℓ)k
k!
(−1)ℓ−i = (−1)ℓ−i
(b+ 1 + i− ℓ)m
m!
,
(by the Chu–Vandermonde sum) which leads to
π−1
ℓ
{
ℓ∑
i=0
(−1)ℓ−i
(b+ 1 + i− ℓ)m
m!
ei(n∗) + (−1)ℓ+1 (b− ℓ+ 1)m
m!
}
= π−1
ℓ
ℓ∑
i=1
(−1)ℓ−i
(b+ 1 + i− ℓ)m
m!
ei(n∗) =
ℓ−1∑
j=0
(−1)j
(b+ 1− j)m
m!
eℓ−j(n∗)
πℓ
;
(with j = ℓ− i) this is the second formula since
(b+ 1− j)m
m!
=
(b+m− j)!
(b− j)!m!
and
eℓ−j(n∗)
πℓ
= ej
(
1
n1
, . . . ,
1
nℓ
)
. ■
The second formula is more concise than the first one when b is relatively small. For example,
when b = 1 (the isotype [N − 1, 1] and p = m+ 2), the value is p− 1− e1
(
1
n1
, . . . , 1
nℓ
)
; this was
already found in [3, Theorem 5.6].
Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups 17
Another interesting specialization of the first formula is for ni = 1 for all i so that the spherical
function reduces to the character
(
τ =
[
N − b, 1b
])
χτ (g) =
(b+ 1)m−ℓ
(m− ℓ)!
+ (−1)ℓ+1 (b− ℓ+ 1)m
m!
, ℓ ≤ m,
(−1)ℓ+1 (b− ℓ+ 1)m
m!
, m < ℓ ≤ N.
Observe that χτ (g) = 0 when b ≤ ℓ− 1 and m ≥ 1. If N = b+ 1, m = 0, then τ = sign whose
value at an ℓ-cycle is (−1)ℓ+1.
Acknowledgements
The author is grateful to the referees whose careful reading and detailed suggestions helped to
improve this paper.
References
[1] Ceccherini-Silberstein T., Scarabotti F., Tolli F., Harmonic analysis on finite groups, Cambridge Stud. Adv.
Math., Vol. 108, Cambridge University Press, Cambridge, 2008.
[2] Chalykh O., Dunkl and Cherednik operators, in Encyclopedia of Mathematical Physics, Vol. 3, Academic
Press, Oxford, 2025, 309–327, arXiv:2409.09005.
[3] Dunkl C., Gorin V., Eigenvalues of Heckman–Polychronakos operators, arXiv:2412.01938.
[4] Heckman G.J., An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103
(1991), 341–350.
[5] Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Math. Monogr., The Clarendon
Press, New York, 1995.
[6] Polychronakos A.P., Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69
(1992), 703–705, arXiv:hep-th/9202057.
https://doi.org/10.1017/CBO9780511619823
https://doi.org/10.1016/B978-0-323-95703-8.00060-4
https://doi.org/10.1016/B978-0-323-95703-8.00060-4
http://arxiv.org/abs/2409.09005
http://arxiv.org/abs/2412.01938
https://doi.org/10.1007/BF01239517
https://doi.org/10.1093/oso/9780198534891.001.0001
https://doi.org/10.1093/oso/9780198534891.001.0001
https://doi.org/10.1103/PhysRevLett.69.703
http://arxiv.org/abs/hep-th/9202057
1 Introduction
2 Spherical functions
3 Coordinate systems and invariant sums of alternating polynomials
4 The case p=b+1
5 The cases p>b+1
5.1 Case p=b+2
5.2 Case p>b+2
6 An equivalent formula
References
|
| id | nasplib_isofts_kiev_ua-123456789-213523 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T09:33:16Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dunkl, Charles F. 2026-02-18T11:24:19Z 2025 Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups. Charles F. Dunkl. SIGMA 21 (2025), 053, 17 pages 1815-0659 2020 Mathematics Subject Classification: 20C30; 43A90; 20B30 arXiv:2503.04547 https://nasplib.isofts.kiev.ua/handle/123456789/213523 https://doi.org/10.3842/SIGMA.2025.053 A Young subgroup of the symmetric group N, the permutation group of {1, 2, …, }, is generated by a subset of the adjacent transpositions {(, +1)∣1 ≤ < }. Such a group is realized as the stabilizer ₙ of a monomial λ (=λ¹₁λ²₂ ⋯ λᴺN) with λ = (ⁿ¹₁, ⁿ²₂, …, ⁿᵖₚ) (meaning ⱼ is repeated ⱼ times, 1 ≤ ≤ , and ₁ > ₂ > ⋯ > ₚ ≥ 0), thus it is isomorphic to the direct product ₙ₁ × ₙ₂ × ⋯ ×ₙₚ. The interval {1, 2, …, } is a union of disjoint sets ⱼ = { ∣ λᵢ = ⱼ}. The orbit of λ under the action of N (by permutation of coordinates) spans a module λ, the representation induced from the identity representation of ₙ. The space λ decomposes into a direct sum of irreducible N-modules. The spherical function is defined for each of these; it is the character of the module averaged over the group ₙ. This paper concerns the value of certain spherical functions evaluated at a cycle that has no more than one entry in each interval ⱼ. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by . Gorin and the author [arXiv:2412:01938]. In particular, the present paper determines the spherical function value for N-modules of hook tableau type, corresponding to Young tableaux of shape [ − , 1ᵇ]. The author is grateful to the referees whose careful reading and detailed suggestions helped to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups Article published earlier |
| spellingShingle | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups Dunkl, Charles F. |
| title | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_full | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_fullStr | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_full_unstemmed | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_short | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_sort | some spherical function values for hook tableaux isotypes and young subgroups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213523 |
| work_keys_str_mv | AT dunklcharlesf somesphericalfunctionvaluesforhooktableauxisotypesandyoungsubgroups |