Vertex operators and the class algebras of symmetric groups

Alain Lascoux and Jean-Yves Thibon

Dedicated to the memory of Sergei Kerov

Abstract

We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of W_{1+∞}, to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski.

1 Introduction

Convolution of central functions, or multiplication in the center of the group algebra of the symmetric group 𝔖_{n} can be realized by means of differential operators acting on symmetric functions. This is due to the existence of the Frobenius map, which sends a permutation σ of cycle type α to the product of power sums p_{α}. Since the power sums are algebraically independent, any linear operator on the space Sym_{n} of homogeneous symmetric functions of degree n can be realized as a differential operator in the variables p_{k}, and the above statement is therefore trivial. More interesting is the existence of infinite order differential operators Φ implementing simultaneously for all symmetric groups the multiplication by families of elements η_{n} in the center Z𝔖_{n} of ℂ𝔖_{n}.

The first example of such an operator has been given by Goulden [4], for the case where η_{n} = C_{(2,1^{n−2})} is the sum of all transpositions in 𝔖_{n}, and similar operators for η_{n} = C_{(ρ,1^{n−r})} (where ρ is a partition of r) have been recently described by Goupil, Poulhalon and Schaeffer [5].

Convolution of central functions is not the only operation which presents stability properties allowing to deal simultaneously with all symmetric groups. This is also the case of pointwise multiplication, which, applied to characters, corresponds to the tensor product of representations. Here, the stability properties are best explained in terms of vertex operators [18, 16], which can also produce stable formulas for outer or inner plethysms, character values or branching multiplicities [2, 17].

One might therefore expect that vertex operators play a role in the simultaneous description of the centers of all symmetric group algebras. A further hint that this should be the case is the observation by Frenkel and Wang [3] that the commutators of Goulden’s operator with the operators α_{−k} = “multiplication by p_{k}” and their adjoints α_{k}, generate a representation of the Virasoro algebra.

Actually, vertex operators arise when one generalizes Goulden’s formula in another direction. Rather than considering the sum of all transpositions in 𝔖_{n} as the conjugacy class C_{(2,1^{n−2})}, one can view it as the sum p_{1}(Ξ_{n}) = ∑_{i=1}^{n} ξ_{i} of the Jucys-Murphy elements, and look for a generating function of the differential operators D_{k} implementing the multiplication by power sums p_{k}(Ξ_{n}) for all n.

One finds that the required generating function has a simple expression in terms of the classical vertex operator Γ(z_{1}, z_{2}) describing the Fock space representations of 𝔤l_{∞} and 𝔤̂l_{∞}. Then, one can see that the brackets [D_{k}, α_{l}] generate a charge 1 representation of the Lie algebra W_{1+∞}, by expressing them in closed form in terms of the standard generators of this algebra (for k = 1, this is the result of Frenkel and Wang).

In Section 4, we remark that our vertex operator provides a new approach to the results of Kerov and Olshanski [11]. What we prove is that the coefficients of the products of power-sums of Jucys-Murphy elements p_{μ}(Ξ_{n}) on the normalized conjugacy classes of [11] are independent of n, which is equivalent to Proposition 3 of [11].

In Section 5, we give an alternative derivation of the operators of [5] in terms of matrix integrals. We start, following Hanlon, Stanley and Stembridge [6], with the observation that the theory of spherical functions on the cone of positive definite Hermitian matrices allows one to write generating functions for connection coefficients as Gaussian integrals over the space of complex N × N matrices, N being sufficiently large (actually, it is the limit N → ∞ which is relevant, and we are in fact considering functional integrals). Then, we combine the generating functions for all n, and we are reduced to the evaluation of similar integrals, but for a modified Gaussian measure, which can be performed by means of Wick’s formula. On the way, we observe that this method of calculation can give a direct combinatorial proof of the generating function of [6], whithout any reference to spherical functions (this answers a question raised in the last section of [6]).

Finally, we observe that the results of Kerov and Olshanski rederived in Section 4 imply that the Goupil-Poulhalon-Schaeffer operators form a linear basis of the commutative subalgebra of U(W_{1+∞}) generated by the D_{k}.

2 Notations and background

2.1 Symmetric functions

We denote by Sym the (abstract) algebra of symmetric functions, with complex coefficients, and by Sym(X) = Sym(x_{1}, …, x_{n}) the algebra of symmetric polynomials in n variables. The homogeneous component of degree k is denoted by Sym_{k}. The scalar product on Sym is the standard one, for which the Schur functions s_{λ} form an orthonormal basis. For f ∈ Sym, D_{f} denotes the adjoint of the operator g ↦ fg.

For A = {a_{1}, a_{2}, … } we denote by σ_{z}(A) = ∏_{i}(1 − za_{i})^{−1} and λ_{z}(A) = σ_{−z}(A)^{−1} the generating series of complete and elementary symmetric functions of A. Other notations for symmetric functions are as in [13].

2.2 The Frobenius characteristic map

The conjugacy class of permutations with cycle type μ is denoted by C_{μ}. A central function f on 𝔖_{n} is identified to the element F = ∑_{σ} f(σ)σ ∈ Z𝔖_{n}. The Frobenius characteristic map is the linear map ch : ℂ𝔖_{n} → Sym_{n} defined by ch (σ) = p_{μ} is σ is of cycle type μ. The structure constants c_{αβ}^{γ} of Z𝔖_{n} are defined by C_{α}C_{β} = ∑_{γ} c_{αβ}^{γ}C_{γ}.

The Frobenius map allows one to define a new product × on each Sym_{n} by ch (F) × ch (G) = ch (F G). We extend it to Sym by setting u × v = 0 if u and v are homogeneous of different degrees. Then, s_{λ} × s_{μ} = \frac{1}{f^{λ}} δ_{λμ}s_{λ}, where f^{λ} is the dimension of the representation λ of 𝔖_{n}. We denote by Γ the comultiplication dual to ×, that is, Γ(s_{λ}) = \frac{1}{f^{λ}} s_{λ} ⊗ s_{λ}.

2.3 Jucys-Murphy elements

The Jucys-Murphy elements of 𝔖_{n} are the n sums of transpositions [7, 14]

ξ_{j} = ξ_{j;n} = ∑_{i<j}(i, j) .  (1)

Note that ξ_{1} is zero, but it is convenient to include it as well. These elements generate a maximal commutative algebra GZ_{n} of ℂ𝔖_{n} (the Gelfand-Zetlin subalgebra), and the center of ℂ𝔖_{n} is Sym(ξ_{1}, …, ξ_{n}). Young’s othogonal idempotents e_{t,t} (t a standard tableau) belong to GZ_{n}, and ξ_{i}e_{t,t} = c_{i}(t)e_{t,t} where c_{i}(t) is the content of the box labelled i in t (the content of the box in row k and column l of a Young diagram is defined as l − k). The multiset of contents of a partition λ is denoted by C(λ) = {c_{□}| □ ∈ λ} (where □ runs over all boxes in the diagram of λ).

Jucys has shown that the elementary symmetric function e_{k} of the ξ_{i} is equal to the sum of all permutations having exactly n − k cycles. One can check that the products

eᾱ = e_{α_{2}}e_{α_{3}} ⋯ e_{α_{r}} , α = (α_{1}, α_{2}, … α_{r}) ⊢ n

form a linear basis of Z𝔖_{n}. For example, for n = 4, a basis is {e_{0}, e_{1}, e_{2}, e_{3}, e_{11} = 2C_{22}+3C_{31}+6C_{1111}}. However, in the sequel, we shall rather work with power sums.

2.4 The Fock space formalism

We will also identify Sym with the infinite wedge space F^{(0)}, spanned by semi-infinite products w = v_{i_{1}} ∧ v_{i_{2}} … (i_{k} ∈ ℤ) such that i_{1} > i_{2} > …, and i_{k} = 1 − k for k >> 0. Such a vector will be denoted by |λ⟩, where the partition λ is defined by λ_{k} = i_{k}+k−1. This space is the basic representation L(Λ_{0}) of the affine Lie algebra 𝔤̂l_{∞} = 𝔄_{∞}, the universal central extension of the Lie algebra of ℤ × ℤ-matrices with a finite number of nonzero diagonals. The generators E_{ij} act in a simple way of the semi-infinite wedges. For i ≠ j, if v_{j} occurs in a wedge w, E_{ij} replaces it with v_{i}, otherwise the result is 0.

For i > 0, E_{ii}w = w if v_{i} occurs in w, and 0 otherwise. For i ≤ 0, the result is 0 if v_{i} occurs in w, and −w otherwise.

The Boson-Fermion correspondence produces differential operators transporting this action on Sym under the linear isomorphism |λ⟩ → s_{λ} (see [8], Chap. 14). More generally, the space F^{(m)} is spanned by wedges such that i_{k} = 1 − k + m for k >> 0. Their direct sum (for m ∈ ℤ) is called the fermionic Fock space, and F^{(m)} is called the charge m sector.

3 A vertex operator for power sums of Jucys-Murphy elements

3.1 A differential operator for 𝔖_{n}

In this section, we will compute the differential operator D^{(n)} acting on Sym_{n} as the generating function

F_{n}(t) = ∑_{k≥1} p_{k}(Ξ_{n}) = ∑_{i=1}^{n}(e^{tξ_{i}} − 1) ,  (2)

that is, for P ∈ Sym_{n}, D^{(n)}P := ch (F_{n}(t)) × P.

We know that the eigenvalue of p_{k}(Ξ_{n}) on the central idempotent e_{λ} is p_{k}(C(λ)). Therefore, the eigenvalue of D^{(n)} on s_{λ} is

∑_{□∈λ} (e^{tc_{□}} − 1) = ∑_{□∈λ}(q^{c_{□}} − 1)  (3)

if we set q = e^{t}. This sum is easily evaluated in terms of the parts of λ:

Lemma 3.1 Let λ be a non zero partition of length at most n. Then,

∑_{□∈λ} q^{c_{□}} = \frac{q}{q − 1} ∑_{i=1}^{n} (q^{λ_{i}−i} − q^{−i}) .

Proof – The contents of λ are the numbers −i + 1, −i + 2, …, λ_{i} − 1 for i = 1, …, ℓ(λ).

Therefore,

∑_{□∈λ}(q^{c_{□}} − 1) = \frac{q}{q − 1} ∑_{i=1}^{n} (q^{λ_{i}−i} − q^{−i}) − ∑_{i=1}^{n} λ_{i} .  (4)

Our first task is to express the operator induced by D^{(n)} on the space of symmetric polynomials Sym(x_{1}, …, x_{n}) in terms of the variables x_{i}. We set

Δ_{n} = ∏_{i<j}(x_{i} − x_{j}) and □_{n} = (x_{1}x_{2} ⋯ x_{n})^{n} .  (5)

Let D_{i} = x_{i} \frac{∂}{∂x_{i}}. We have

Lemma 3.2

\frac{□_{n}}{Δ_{n}} (∑_{i=1}^{n} q^{D_{i}}) \frac{Δ_{n}}{□_{n}} · s_{λ} = (∑_{i=1}^{n} q^{λ_{i}−i}) s_{λ} .

Proof – Multiplication of s_{λ} by Δ_{n}□_{n}^{−1} results in the determinant det(x_{j}^{λ_{i}−i}). Applying ∑_{i} q^{D_{i}} to this determinant amounts to apply the one-variable operator q^{D} to each row of the determinant, and then take the sum. This produces the same result as applying the operator to each column succesively, since both expressions are equal to the coefficient of ϵ in det((1+ϵq^{D})(x_{j}^{λ_{i}−i})).

Therefore, the operator

\frac{q}{q − 1} \frac{□_{n}}{Δ_{n}} (∑_{i=1}^{n} q^{D_{i}} − q^{−i}) \frac{Δ_{n}}{□_{n}} − ∑_{i=1}^{n} D_{i}  (6)

has the same eigenvalues as D^{(n)} on Schur functions s_{λ} in n variables, and must therefore coincide with it. We can now let n → ∞, and see that D^{(n)} is the restriction to Sym_{n} of the well-defined limit

D = lim_{n→∞} D^{(n)} .  (7)

Hence, all symmetric groups can be dealt with simultaneously by the single operator D.

3.2 Bosonization

The next step is to express D in terms of the power sums. To avoid confusion, we reserve the letter X for finite sets of variables, and introduce an infinite alphabet A as argument of our symmetric functions. So, we want to express the action of D on Sym(A) in terms of the operators

α_{−k} = p_{k}(A) , α_{k} = α_{−k}^{†} = D_{p_{k}} = k \frac{∂}{∂p_{k}(A)} (k ≥ 1) .  (8)

This procedure is called bosonization in the physics literature (see, e.g., [1]), for the α_{k} satisfy the commutation relations

[α_{j}, α_{k}] = jδ_{j,−k}  (9)

of the modes of a free boson field (a Heisenberg algebra).

To compute the bosonization of D, we have to calculate the bi-symmetric kernel

K(X, A) = λ_{−1}(XA)D^{(n)}σ_{1}(XA)  (10)

where D^{(n)} acts on functions of X = {x_{1}, …, x_{n}}, and to express it in the form

K(X, A) = ∑_{μ,ν} k_{μν}p_{μ}(X)p_{ν}(A) .  (11)

Then, we will have

D = ∑_{μ,ν} k_{μν}p_{μ}(A)D_{p_{ν}(A)} .  (12)

Indeed, writing ⟨ , ⟩ for the scalar product of Sym(A), we have f(X) = ⟨σ_{1}(XA), f(A)⟩, so that

D^{(n)}f(X) = ⟨D^{(n)}σ_{1}(XA), f(A)⟩ = ⟨1, D_{D^{(n)}σ_{1}(XA)}f(A)⟩ = ⟨σ_{1}(XA), Df(A)⟩ .

Let ∇_{i} be the partial q-derivative with respect to x_{i}, i.e.

∇_{i} = \frac{q^{D_{i}} − 1}{(q − 1)x_{i}} .

We have, for any function f(X) = f(x_{1}, …, x_{n}),

\frac{q}{q − 1} (∑_{i=1}^{n} q^{λ_{i}−i}) f(X) = (∑_{i=1}^{n} ∇_{i}x_{i}) f(X) + (∑_{i=1}^{n}[i]_{1/q}) f(X) .

To apply this to f(X) = Δ_{n}σ_{1}(XA)/□_{n}, we note that

(∑_{i=1}^{n} ∇_{i}x_{i}) \frac{Δ_{n}}{□_{n}} = [\frac{n}{1 − q} + \frac{q(1 − q^{−n})}{(1 − q)^{2}}] \frac{Δ_{n}}{□_{n}}  (13)

and

(∑_{i=1}^{n} ∇_{i}x_{i}) \frac{Δ_{n}}{□_{n}} σ_{1}(XA) = (∑_{i=1}^{n} ∇_{i}x_{i} \frac{Δ_{n}}{□_{n}} ) σ_{1}(XA) + ∑_{i=1}^{n} \frac{qx_{i}Δ^{(i)}}{q^{n}□_{n}} ∇_{i}σ_{1}(XA)

(14)

where

Δ^{(i)} = q^{D_{i}}Δ_{n} = Δ_{n}A_{i}(X; q) , A_{i}(X; q) = ∏_{j≠i} \frac{qx_{i} − x_{j}}{x_{i} − x_{j}} .

Hence, setting D̄^{(n)} = D^{(n)} + E^{(n)}, where E^{(n)} is the Euler operator, we have

D̄^{(n)}σ_{1}(XA) = [(\frac{n}{1 − q} + \frac{q(1 − q^{−n})}{(1 − q)^{2}}i) + ∑_{i=1}^{n}[i]_{1/q}] σ_{1}(XA)

+q^{1−n} ∑_{i=1}^{n} x_{i}A_{i}(X; q)∇_{i}σ_{1}(XA)

= q^{−n} ∑_{i=1}^{n} A_{i}(X; q)\frac{qx_{i}}{(q − 1)x_{i}} (\frac{σ_{qx_{i}}(A)σ_{1}(XA)}{σ_{x_{i}}(A)} − σ_{1}(XA))

= σ_{1}(XA)\frac{q^{−n}}{q − 1} ∑_{i=1}^{n} A_{i}(X; q)(σ_{qx_{i}}(A)λ_{−x_{i}}(A) − 1)

= σ_{1}(XA)\frac{q^{−n}}{q − 1} ∑_{i=1}^{n} A_{i}(X; q)(σ_{x_{i}}((q − 1)A) − 1)

= σ_{1}(XA)\frac{q^{−n}}{q − 1} ∑_{m≥1} h_{m}((q − 1)A) ∑_{i=1}^{n} A_{i}(X; q)x_{i}^{m}

= σ_{1}(XA)\frac{q^{−n}}{q − 1} ∑_{m≥1} h_{m}((q − 1)A)q^{n} \frac{h_{m}((1 − q^{−1})A)}{1 − q^{−1}} .

Rewriting this expression in a more symmetric form, we obtain the kernel of D̄ = D + E

K̄(X; A) = \frac{q}{(q − 1)^{2}} ∑_{m≥1} q^{−m}h_{m}((q − 1)A)h_{m}((q − 1)X) .  (15)

The bosonization of the Euler operator being obviously E = ∑_{k≥1} p_{k}D_{p_{k}}, we have Proposition 3.3 The differential operator corresponding to ∑_{i≥1}(q^{ξ_{i}} −1) is

D = \frac{q}{(q − 1)^{2}} ∑_{m≥1} q^{−m}h_{m}((q − 1)A)D_{h_{m}((q−1)A)} − E .

On this expression, it is clear that D can be written

D = \frac{V_{0} − 1}{(q − 1)(1 − q^{−1})} − E

where V_{0} is the zero mode of the vertex operator

V (z; q) = σ_{z}((q − 1)A)D_{σ_{1/z}((1−q^{−1})A)}
= exp \{∑_{k≥1}(q^{k} − 1)p_{k} \frac{z^{k}}{k} \} exp \{∑_{l≥1}(1 − q^{−l})z^{−l} \frac{∂}{∂p_{l}} \}

= : exp \{∑_{k≠0} \frac{(1 − q^{−k})z^{−k}}{k} α_{k}\} :

= ∑_{m=−∞}^{∞} V_{m}(q)z^{−m} .

This operator satisfies the commutation relations

[V, α_{k}] = z^{k}(1 − q^{k})V (k ≠ 0)  (16)

so that

[V_{l}, α_{k}] = (1 − q^{k})V_{k+l} (k, l ∈_{Z}, k ≠ 0).  (17)

In particular,

V_{k} = (1 − q^{k})^{−1}[V_{0}, α_{k}] (k ≠ 0)  (18)

that is, all the modes are generated by the action of the bosonic operators on V_{0}.

3.3 Class algebras and infinite dimensional Lie algebras

The vertex operator V (z; q) is well-known to be related to the Fock space representations of various infinite dimensional Lie algebras (see e.g., [8], Corollary 14.10). In the notation of [8], V (z; q) = Γ(qz, z), and if r̂_{m} denotes the representation of 𝔤̂l_{∞} in the charge m sector F^{(m)} of the fermionic Fock space F, one has in particular

W (z; q) = \frac{1}{1 − q^{−1}} (V (z; q) − 1) = ∑_{i,j∈ℤ}q^{i}z^{j}r̂_{0}(E_{i,i−j}) .  (19)

Write W (z; q) = ∑_{k} W_{k}(q)z^{−k}, so that

W_{k}(q) = r̂_{0} (∑_{i∈ℤ}q^{i}E_{i,i+k}) .  (20)

The commutation relations between the operators W_{k}(q) are easily determined from the defining relations of 𝔤̂l_{∞}, which read

[E_{ij}, E_{kl}] = δ_{jk}E_{il} − δ_{li}E_{ki} + Ψ(E_{ij}, E_{kl})c  (21)

where c is the central charge, and Ψ is the 2-cocycle of 𝔤l_{∞} given by

Ψ(E_{ij}, E_{ji}) = −Ψ(E_{ji}, E_{ij}) = 1 if i ≤ 0, j ≥ 1  (22)

and Ψ(E_{ij}, E_{kl}) = 0 in all other cases. One has r̂_{0}(c) = 1, and a short calculation yields

[W_{k}(a), W_{l}(b)] = (b^{k} − a^{l})W_{k+l}(ab) + δ_{k,−l} \frac{b^{−l} − a^{−k}}{1 − (ab)^{−1}} .  (23)

One recognize that these relations are almost the standard presentation (in generating function form such as in [10], Eq. (2.2.2)) of the Lie algebra usually denoted by D̂ or W_{1+∞}, the universal central extension of the Lie algebra D of all differential operators on the circle. The generators of D are the z^{k}D^{n}, where D = z∂_{z}, and the corresponding elements of the central extension D̂ are denoted by L_{k}^{n}. The cocycle of the central extension is given by

Φ(z^{k}f(D), z^{l}g(D)) = ∑_{j≥1} kf(−j)g(k − j) if k = −l ≥ 0  (24)

and is 0 in all other cases. With this at hand, we see that the operators

T_{k}(q) = −q^{−1}W_{k}(q)  (25)

satisfy

[T_{k}(a), T_{l}(b)] = (a^{l} − b^{k})T_{k+l}(ab) + δ_{k,−l} \frac{a^{−k} − b^{−l}}{1 − ab}  (26)

which is exactly Eq. (2.2.2) of [10] with C = 1. Therefore, the coefficients T_{k,n} defined by

T_{k}(e^{t}) = ∑_{n≥0} \frac{t^{n}}{n!}T_{k,n}  (27)

are the images of the L_{n}^{k} in a representation R_{0} of charge 1.

Now, our differential operator D reads

D = \frac{−1}{1 − q^{−1}} T_{0}(e^{t}) − E = \frac{1}{t} \frac{−t}{e^{−t} − 1} ∑_{l≥0} \frac{t^{l}}{l!}T_{0l} − E

= ∑_{n≥1} \frac{t^{n}}{n!} ∑_{k=0}^{n}(−1)^{k−1}\binom{n}{k}B_{k} \frac{T_{0,n+1−k}}{n + 1 − k}

= ∑_{n≥1} \frac{t^{n}}{n!}D_{n} .

We have T_{00} = 0, T_{01} = −E, and Goulden’s operator is D_{1} = −\frac{1}{2}(T_{02} + T_{01}). Also, T_{k,1} = α_{k}, T_{k,2} = 2L_{k} where the L_{k} are the charge 1 Virasoro operators considered in [3]. Since [T_{0}(q), α_{k}] = (1 − q^{k})T_{k}(q), we find that the result of [3] stating that the commutators [D_{1}, α_{k}] generate a Virasoro algebra can be extended as follows:

Proposition 3.4 The commutators [D_{j}, α_{k}] generate a charge 1 representation of W_{1+∞}.

3.4 Interpretation of the Virasoro operators

It would be of interest to have interpretations of the other generators T_{ij} (i ≠ 0) in terms of natural operations on Z𝔖 = ⨁_{n≥0} Z𝔖_{n}. As a step in this direction, we can propose such an interpretation for the positive part of the Virasoro algebra.

Let, for k ≥ 1

d′_{k} = ∑_{j≥0} p_{j}D_{p_{j+k}} = ∑_{j≥0} α_{−j}α_{j+k} ,  (28)

d′′_{k} = \frac{1}{2} ∑_{{1≤i,j \atop i+j=k}} α_{i}α_{j} ,  (29)
d_{k} = d′_{k} + d′′_{k} ,  (30)

where p_{0} = 1.

Let also δ_{k}, δ′_{k} and δ′′_{k} be the linear maps ℂ𝔖_{n} → ℂ𝔖_{n−k} defined on permutations by

δ′_{k}(σ) = \frac{1}{(l − 1)(l − 2) ⋯ (l − k)}σ^{(k)} ,  (31)
δ′′_{k}(σ) = \frac{1}{2k!} ∑_{{1≤i,j \atop i+j=k}} ij σ^{(i,j)} ,  (32)
δ_{k}(σ) = δ′_{k}(σ) + δ′′_{k}(σ) ,  (33)

where σ^{(k)} (resp. σ^{(i,j)}) are defined to be 0 if n, n − 1, …, n − k + 1 do not belong to the same cycle of length l (resp. do not constitute two cycles of lengths i, j), and otherwise, σ^{(k)} and σ^{(i,j)} are the permutations whose cycle decomposition is obtained by erasing n, n − 1, …, n − k + 1 in the cycle decomposition of σ.

Proposition 3.5 For u ∈ Z𝔖_{n}, one has

\frac{1}{(n)_{k}} ch (δ_{k}u) = d_{k}ch (u) ,
\frac{1}{(n)_{k}} ch (δ′_{k}u) = d′_{k}ch (u) .

Proof – A direct calculation. The numerical factors account for the orders of conjugacy classes, and could have been suppressed by using instead the basis

b_{μ} = \frac{1}{n!} ∑_{τ∈𝔖_{n}} τ^{−1}στ

where σ is any permutation of cycle type μ.

The operators d_{i} are the images of the generators L_{i} of the Virasoro subalgebra of W_{1+∞} under the above representation, while the d′_{i} correspond to the Witt algebra. Therefore, L_{1} corresponds to the map considered in [12], and L_{2} amounts to erasing n and n − 1 if they are both in the same cycle, or both fixed points. Similarly, L_{3} erases n, n − 1, n − 2 if they are in the same cycle, or constitute two cycles of lengths 1 and 2.

4 A stability property

The previous result can be used to express the power-sums of Jucys-Murphy elements as linear combinations of conjugacy classes. Indeed, for fixed n, the generating function

J_{n}(t) = ∑_{k≥0} ch (p_{k}(Ξ_{n}))\frac{t^{k}}{k!} = ∑_{k≥0} J_{n}^{k} \frac{t^{k}}{k!}

is equal to the constant term of \frac{1}{(q−1)(1−q^{−1})}(V (z; q) − 1)p_{1}^{n}, that is, to

\frac{1}{(q − 1)(1 − q^{−1})} [∑_{k=0}^{n} \binom{n}{k}p_{1}^{n−k}h_{k}((q − 1)A)(1 − q^{−1})^{k}]

which has to be expanded with q = e^{t} and

h_{k}((q − 1)A) = ∑_{μ⊢k} [∏_{i}(e^{μ_{i}t} − 1)] z_{μ}^{−1}p_{μ}(A) .

This is best accomplished by means of a generating function. We have

J (t) = ∑_{n≥0} \frac{1}{n!}J_{n}(t) = [z^{0}]\frac{(V (z; q) − 1)e^{p_{1}}}{(q − 1)(1 − q^{−1})}

= e^{p_{1}} ∑_{k≥1} \frac{(1 − q^{−1})^{k−1}}{k!} \frac{h_{k}((q − 1)A}{q − 1}

= e^{p_{1}} ∑_{k≥1} \frac{(1 − q^{−1})^{k−1}}{k!} ∑_{κ⊢k} \frac{p_{κ}(q − 1)}{q − 1} \frac{p_{κ}(A)}{z_{κ}} .

For κ ⊢ k ≥ 1, let

ϕ_{κ}(t) = \frac{(1 − q^{−1})^{k−1}}{k!z_{κ}} \frac{p_{κ}(q − 1)}{q − 1}|_{q=e^{t}}  (34)

so that if κ = (1^{k_{1}}2^{k_{2}} ⋯),

ϕ_{κ}(t) = \frac{t^{|κ|+ℓ(κ)−2}}{k!k_{1}!k_{2}! ⋯} (1 + O(t)) ,  (35)

and

J (t) = e^{p_{1}} ∑_{|κ|≥1} ϕ_{κ}(t)p_{κ}(A)

= ∑_{n≥0} \frac{1}{n!} ∑_{k=1}^{n} ϕ_{κ}(t)a_{κ;n}

where we have set a_{κ;n} = (n)_{k}p_{κ,1^{n−k}} = ch (a_{κ;n}), where a_{κ;n} are the normalized conjugacy classes defined in [11]. Hence, if

ϕ_{κ}(t) = ∑_{m≥|κ|+ℓ(κ)−2} ϕ_{κ;m} \frac{t^{m}}{m!}  (36)

we obtain

J_{n}^{m} = ch (p_{m}(Ξ_{n})) = ∑_{k=1}^{m+1} ∑_{{κ⊢k \atop ℓ(κ)≤m−k+2}} ϕ_{κ;m}a_{κ;n} .  (37)

Observe that the coefficients are independent of n. Actually, this is a special case of a result of Kerov and Olshanski [11], which is equivalent to the existence of a similar n-independent expansion of all products of power sums p_{μ}(Ξ_{n}) as linear combinations of the a_{κ;n}. This more general result can also be obtained by the same method, but the expressions of the coefficients ϕ_{κ;μ} become more cumbersome. Instead, we observe that if we can prove that the coefficients of the expansion p_{m}((Ξ_{n}) × a_{κ;n} on the basis a_{ν;n} are independent of n, the general result will follow by induction.

To prove this, consider the generating function

G(t; A, B) = ∑_{n≥0} ∑_{m≥0} ∑_{κ} \frac{1}{n!m!}t^{m}J_{n}^{m} × a_{κ;n}(A)\frac{p_{κ}(B)}{z_{κ}} .  (38)

A calculation similar to the previous one (which is the case B = 0) shows that

G(t; A, B) = e^{p_{1}(A)}σ_{1}(AB) ∑_{r≥1} \frac{h_{r}((q − 1)A)h_{r}((1 − q^{−1})(B + E))}{(q − 1)(1 − q^{−1})} (39)

where symmetric functions of the “exponential alphabet” E are defined by σ_{t}(E) = e^{t} (i.e. p_{1}(E) = 1 and p_{k}(E) = 0 for k > 1). On this expression, it is clear that the coefficient of \frac{t^{m}}{m!} \frac{p_{κ}(B)}{z_{κ}} in e^{−p_{1}(A)}G(t; A, B) is a polynomial ∑_{μ} d_{κ;m}^{μ}p_{μ}(A), so that

p_{m}(Ξ_{n})a_{κ;n} = ∑_{μ} d_{κ;m}^{μ}a_{μ;n}  (40)

the coefficients being independent of n, as required.

Here is a table of n-independent expressions of the first power-sums of Jucys-Murphy elements in terms of normalized conjugacy classes.

p_{1}(Ξ) = \frac{1}{2}a_{2}
p_{2}(Ξ) = \frac{1}{3} a_{3} + \frac{1}{2} a_{11}
p_{3}(Ξ) = \frac{1}{4} a_{4} + a_{21} + \frac{1}{2} a_{2}
p_{4}(Ξ) = \frac{1}{5} a_{5} + \frac{1}{2} a_{22} + a_{31} + \frac{2}{3} a_{111} + \frac{5}{3} a_{3} + \frac{1}{2} a_{11}
p_{5}(Ξ) = \frac{1}{6} a_{6} + a_{32} + a_{41} + \frac{5}{2} a_{211} + \frac{15}{4} a_{4} + 5 a_{21} + \frac{1}{2} a_{2}
p_{6}(Ξ) = \frac{1}{7} a_{7} + \frac{1}{2} a_{33} + a_{42} + a_{51} + 3 a_{221} + 3 a_{311} + 7 a_{5}

+\frac{5}{4} a_{1111} + \frac{25}{4} a_{22} + 15 a_{31} + \frac{10}{3} a_{111} + 7 a_{3} + \frac{1}{2} a_{11}

One obtains the expression of each p_{m}(Ξ_{n}) from the table by substituting [(n − k)!]^{−1}z_{κ,1^{n−k}}C_{κ,1^{n−k}} to a_{κ}, κ ⊢ k. For example,

p_{2}(Ξ_{n}) = \frac{1}{3} \frac{3 · (n − 3)!}{(n − 3)!}C_{3,1^{n−3}} + \frac{1}{2} \frac{n!}{(n − 2)!}C_{1,1,1^{n−2}}
= C_{3,1^{n−3}} + \binom{n}{2}C_{1^{n}} .

5 A matrix integral approach

In this section, we will express generating functions for the coproducts of the elements a_{ρ;n} as certain Gaussian integrals over the space of N × N complex matrices. Evaluating these integrals by Wick’s formula, we obtain as a byproduct a new derivation of the differential operators of [5].

The zonal spherical functions of the Gelfand pair (GL(N, ℂ), U(N)) are known to be expressible in terms of Schur functions (see [13], Chap. VII, Sec. 5):

Ω_{λ}(Z) = \frac{s_{λ}(ZZ^{∗})}{s_{λ}(N)} .

As a consequence, we have closed form evaluations of the matrix integrals

∫_{M_{N} (ℂ)} s_{λ}(AZBZ^{∗})dν(Z) = 2^{|λ|}h(λ)s_{λ}(A)s_{λ}(B)  (41)

where h(λ) is the product of the hook-lengths of λ, A and B are arbitrary Hermitian matrices, and dν is the Gaussian probability measure

dν(Z) = (2π)^{−N^{2}}e^{− \frac{1}{2} tr (ZZ^{∗})}dZ , dZ = ∏_{k,l=1}^{N} dx_{kl}dy_{kl} , z_{kl} = x_{kl} + iy_{kl} .

(42)

If |λ| = n, the right-hand side of (41) is

2^{n}n! \frac{s_{λ}(A)s_{λ}(B)}{f_{λ}} = 2^{n}n! Γ(s_{λ})(A ⊗ B)

where Γ is the comultiplication dual to the ×-product, induced on Sym by the convolution of central functions, and elements of Sym ⊗ Sym are interpreted as functions of tensor product of (square) matrices. Therefore, denoting by u_{λ}^{∗} the adjoint of a basis u_{λ},

Γ(p_{λ}) = ∑_{α,β}⟨Γ(p_{λ}), p_{α}^{∗} ⊗ p_{β}^{∗}⟩p_{α} ⊗ p_{β}

= \frac{1}{(n!)^{2}} ∑_{α,β}⟨p_{λ}, C_{α} × C_{β}⟩p_{α} ⊗ p_{β}

= \frac{1}{n!} ∑_{α,β} c_{αβ}^{λ}p_{α} ⊗ p_{β}

(where we have set C_{α} = ch C_{α}), so that [6]

∫_{M_{N} (ℂ)} p_{λ}(AZBZ^{∗})dν(Z) = 2^{n} ∑_{α,β⊢n} c_{αβ}^{λ}p_{α}(A)p_{β}(B) .  (43)

We will now form generating functions for the coproducts. Let ρ be a partition of r. Then,

∫_{M_{N} (ℂ)} p_{ρ}(AZBZ^{∗})\frac{[p_{1}(AZBZ^{∗})]^{n−r}}{2^{n−r}} dν(Z) = 2^{r} ∑_{α,β⊢r} c_{αβ}^{ρ1^{n−r}}p_{α}(A)p_{β}(B)
= 2^{r}n! Γ(p_{ρ1^{n−r}}) .

Therefore, the coproduct of the element a_{ρ} = ∑_{n} a_{ρ;n} is given by

Γ (∑_{n≥r}(n)_{r}p_{ρ1^{n−r}}) = ∫_{M_{N} (ℂ)} p_{ρ}(AZBZ^{∗})e^{\frac{1}{2} p_{1}(AZBZ^{∗})}dν(Z) (44)

= ∫_{M_{N} (ℂ)} p_{ρ}(AZBZ^{∗})dμ(Z)  (45)

where

dμ(Z) = dμ_{A,B}(Z) = π^{N^{2}}e^{− \frac{1}{2} tr (ZZ^{∗}−AZBZ^{∗})}dZ  (46)

is again a Gaussian measure if we assume that the eigenvalues of A and B are < 1. Indeed, one can assume that A = diag (a_{i}), B = diag (b_{i}), and in this case, tr (ZZ^{∗} − AZBZ^{∗}) = ∑_{i,j}(1 − a_{i}b_{j})|z_{ij}|^{2}.

The total mass of dμ is

Z = ∫_{M_{N} (ℂ)} dμ(Z) = ∏_{i,j} \frac{1}{1 − a_{i}b_{j}} .  (47)

For a function f on M_{N}(ℂ), let

⟨f⟩ = \frac{1}{Z} ∫_{M_{N} (ℂ)} f(Z)dμ(Z)  (48)

denote its expectation value. Since dμ is Gaussian, we can make use of Wick’s formula, which in this contexts says the following: if f_{1}, f_{2}, …, f_{m} are ℝ-linear forms on M_{N}(ℂ), we have

⟨f_{1} ⋯ f_{2k−1}⟩ = 0  (49)
⟨f_{1} ⋯ f_{2k}⟩ = Hf (⟨f_{i}, f_{j}⟩)  (50)

where the Hafnian of the matrix (⟨f_{i}, f_{j}⟩) is defined by

Hf (⟨f_{i}, f_{j}⟩)_{1≤i,j≤2k} = ∑ ∏_{i=1}^{k}⟨f_{l_{i}}, f_{m_{j}}⟩  (51)

the sum being taken over all pairs of k-uples L = (l_{1} < l_{2} < ⋯ < l_{k}) and M such that l_{i} < m_{i} and L ∪ M = [1, 2k].

In the case at hand, the “propagators” are given by

⟨z_{ij}z_{kl}⟩ = 0 ,  (52)
⟨z_{ij}^{∗}z_{kl}^{∗}⟩ = 0 ,  (53)
⟨z_{ij}z_{kl}^{∗}⟩ = δ_{il}δ_{jk} \frac{2}{1 − a_{i}b_{j}} .  (54)

Let M = AZBZ^{∗}, and let σ be the following permutation of cycle type ρ

σ = (12 ⋯ ρ_{1})(ρ_{1} + 1, ρ_{1} + 2, ⋯, ρ_{1} + ρ_{2}) ⋯ (⋯ r) .  (55)

Then, our generating function reads

⟨p_{ρ}(M)⟩ = ∑_{i_{1},...,i_{r}} M_{i_{1},i_{σ(1)}}M_{i_{2},i_{σ(2)}} ⋯ M_{i_{r},i_{σ(r)}}
= ∑_{{i_{1},...,ir \atop j_{1},...,jr}} a_{i_{1}}b_{j_{1}}a_{i_{2}}b_{j_{2}} ⋯ a_{i_{r}}b_{j_{r}}⟨z_{i_{1}j_{1}}z_{i_{2}j_{2}} ⋯ z_{i_{r}j_{r}}z_{i_{σ(1)}j_{1}}^{∗} ⋯ z_{i_{σ(r)}j_{r}}^{∗}⟩

= ∑_{I,J} a_{I}b_{J} ∑_{τ∈𝔖_{r}} ∏_{k=1}^{r} δ_{i_{k},i_{στ(k)}}δ_{j_{k},j_{τ(k)}} \frac{2}{1 − a_{i_{k}}b_{j_{k}}}

= 2^{r} ∑_{I,J} ∑_{τ∈𝔖_{r}} G_{I,J} ∏_{k=1}^{r} δ_{i_{k},i_{στ(k)}}δ_{j_{k},j_{τ(k)}}

where we have set G_{i,j} = a_{i}b_{j}(1 − a_{i}b_{j})^{−1} and G_{I,J} = ∏_{k} G_{i_{k},j_{k}} Let us regard the multindices I, J as functions {1, 2, …, r} → ℕ^{∗}. Then, the above product of Kronecker deltas is zero unless I is constant on the orbits of στ, and J is constant on the orbits of τ. To express the final result, we introduce the following notation. Given a permutation τ ∈ 𝔖_{r} and a vector L = (l_{1}, l_{2}, …, l_{r}) of positive integers, let

p_{L}^{τ} = ∏_{k} ∏_{γ∈ k−Cycles(τ)} p_{l_{γ_{1}}+l_{γ_{2}}+···l_{γ_{k}}}  (56)

product on all k-cycles γ = (γ_{1}, …, γ_{k}) of τ. Then,

Proposition 5.1 As a symmetric function of the eigenvalues of A and B,

⟨p_{ρ}(AZBZ^{∗}⟩ = 2^{r} ∑_{l_{1},...,l_{r}≥1} ∑_{τ∈𝔖_{r}} p_{L}^{στ}(A)p_{L}^{τ}(B) .

where σ is defined in (55).

Note that the above calculation provides an answer to a question raised at the end of [6], namely, to find a direct combinatorial proof of (43). Indeed, to obtain the expectations with respect to dν rather than dμ, one just has to replace the propagators by G_{i,j} = a_{i}b_{j}/2, in which case the sum over L disappears (the only remaining term being for L = (1, 1, …, 1)) and one finds exactly (43).

Now, if ρ is a reduced partition (no part equal to 1),

σ_{1}(A ⊗ B)⟨p_{ρ}(AZBZ^{∗})⟩ = 2^{r}z_{ρ} ∑_{n≥r} Γ(C_{ρ,1^{n−r}})(A ⊗ B)  (57)

and in general, the ×-multiplication by any symmetric function F can be implemented by a differential operator as soon as its coproduct is known in the form

Γ(F) = δσ_{1} ∑ f_{αβ}p_{α} ⊗ p_{β} ,  (58)

where δ is the comultiplication defined by δ(p_{μ}) = p_{μ} ⊗ p_{μ}. Indeed, for any symmetric functions G, H,

⟨F × G, H⟩ = ⟨F, H × G⟩ = ⟨Γ(F), H ⊗ G⟩
= ∑_{αβ} f_{αβ}⟨δσ_{1}p_{α} ⊗ p_{β}, H ⊗ G⟩
= ∑_{αβ} f_{αβ}⟨δσ_{1}, D_{p_{α}}H ⊗ D_{p_{β}}G⟩
= ∑_{αβ} f_{αβ}⟨σ_{1}, D_{p_{α}}H ⊗ D_{p_{β}}G⟩
= ∑_{αβ} f_{αβ}⟨D_{p_{β}}G, D_{p_{α}}H⟩

= ⟨∑_{αβ} f_{αβ}p_{α}D_{p_{β}}G, H⟩ .

Hence, we recover the result of [5], proving conjectures of Katriel [9]:

Proposition 5.2 Let ρ be a reduced partition of r. For any homogeneous symmetric function G of degree n ≥ r,

C_{ρ,1^{n−r}} × G = H_{ρ}(G)

where H_{ρ} is the differential operator

H_{ρ} = \frac{1}{z_{ρ}} ∑_{l_{1},...,l_{r}≥1} ∑_{τ∈𝔖_{r}} p_{L}^{στ}D_{p_{L}^{τ}} .

6 Final remarks

Recall that p_{ν} ×s_{λ} = (χ_{ν}^{λ}/f^{λ})s_{λ}, where f^{λ} is the number of standard tableaux of shape λ. Kerov and Olshanski have shown that for a partition ρ of r, the function

f_{ρ}(λ) = (n)_{r} \frac{χ_{ρ,1^{n−r}}^{λ}}{f^{λ}}  (59)

is a polynomial in the “shifted power-sums”

p̃_{k}(λ) = ∑_{i≥1}(λ_{i} − i)^{k} − (−i)^{k}

which are the eigenvalues on |λ⟩ = s_{λ} of the operators

P_{k} = r̂_{0} (∑_{i∈ℤ}i^{k}E_{ii})

of the Fock space representation of 𝔤̂l_{∞}. (We have proved an equivalent result in Section 4.)

A first consequence of this result is that the elements a_{ρ;n} = ch (a_{ρ;n}) (a_{ρ;n} ∈ Z𝔖_{n} are the normalized conjugacy classes of [11]) have structure constants independent of n: there exist nonnegative integers g_{αβ}^{γ} such that

a_{α;n} × a_{β;n} = ∑_{γ} g_{αβ}^{γ}a_{γ;n}  (60)

for all n. Therefore, the operators A_{ρ} implementing simultaneously the multiplication by all a_{ρ;n} also satisfy

A_{α}A_{β} = ∑_{γ} g_{αβ}^{γ}A_{γ}  (61)

so that they form a linear basis of commutative subalgebra of U(𝔤̂l_{∞}). When ρ is reduced, A_{ρ} = z_{ρ}H_{ρ}.

A second consequence is that these operators actually belong to the image of U(W_{1+∞}) under the representation R_{0} of Section 3. Indeed, (59) show that A_{ρ} is a polynomial in the commuting operators P_{k}, which are related to the D_{k} of the previous section by

P_{n} = ∑_{k=0}^{n−1} \binom{n}{k}D_{k}  (62)

where we have set D_{0} = E.

Kerov and Olshanski also identified the algebra of the a_{ρ;n} to an algebra of differential operators A_{ρ;N} living in the center of U(𝔤l_{N}) for N ≥ n. Since these operators commute with the adjoint representation, they preserve the space of functions on GL_{N}(ℂ) which are symmetric functions of the eigenvalues. The previous considerations show then that if we set

p̃_{k}(D) = ∑_{i=1}^{N} D_{i}^{k} − (−i)^{k}

and f_{ρ}(λ) = ∑_{ν} k_{ρν}p̃_{ν}(λ), the restriction of A_{ρ;N} to invariant functions is given by

A_{ρ;N} = \frac{□_{N}}{Δ_{N}} ∑_{ν} k_{ρν}p̃_{ν}(D)\frac{Δ_{N}}{□_{N}} .  (63)

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Institut Gaspard Monge, Université de Marne-la-Vallée, 77454 Marne-la-Vallée cedex, FRANCE