% opc8.tex 6 fev 01 \documentclass[12pt,twoside]{article} \pagestyle{myheadings} \markboth{A. Lascoux and J.-Y. Thibon}{Vertex operators and class algebras} \usepackage{amssymb} \newtheorem{example}{Example}[section] \newtheorem{note}[example]{Note} \newtheorem{theorem}[example]{Theorem} \newtheorem{exercise}{Exercise}[section] \newtheorem{corollary}[example]{Corollary} \newtheorem{conjecture}[example]{Conjecture} \newtheorem{definition}[example]{Definition} \newtheorem{proposition}[example]{Proposition} \newtheorem{algorithm}[example]{Algorithm} \newtheorem{lemma}[example]{Lemma} \def\Proof{\noindent \it Proof -- \rm} \def\qed{\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm} \bigskip} %\def\qed{\hspace{3.5mm} \hfill \vbox{\hrule height 3pt depth 2 pt width 2mm}} \def\bo{{\small \,\Box}} \def\DD{{\sf D}} \def\EE{{\sf E}} \def\<{\langle} \def\>{\rangle} \def\R{{\mathbb R}} \def\C{{\mathbb C}} \def\Z{{\mathbb Z}} \def\N{{\mathbb N}} \def\tr{{\rm tr\,}} \def\gl{{\frak gl}} \def\glchap{\widehat\gl} \def\diag{{\rm diag\,}} \def\Hf{{\rm Hf\,}} \def\SG{{\mathfrak S}} \def\H{{\sf H}} \def\t{{\bf t}} \def\A{{\sf A}} \def\aa{{\sf a}} \def\ch{{\rm ch\,}} \def\CC{{\cal C}} \title{Vertex operators and the class algebras of symmetric groups} \author{Alain {\sc Lascoux} and Jean-Yves {\sc Thibon}} \date{{\footnotesize Dedicated to the memory of Sergei Kerov}} \begin{document} \maketitle \begin{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 ${\cal W}_{1+\infty}$, 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. \end{abstract} \section{Introduction} Convolution of central functions, or multiplication in the center of the group algebra of the symmetric group $\SG_n$ 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 $\sigma$ of cycle type $\alpha$ to the product of power sums $p_\alpha$. 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 $\Phi$ implementing simultaneously for all symmetric groups the multiplication by families of elements $\eta_n$ in the center $Z\SG_n$ of $\C\SG_n$. The first example of such an operator has been given by Goulden \cite{Go}, for the case where $\eta_n=\CC_{(2,1^{n-2})}$ is the sum of all transpositions in $\SG_n$, and similar operators for $\eta_n=\CC_{(\rho,1^{n-r})}$ (where $\rho$ is a partition of $r$) have been recently described by Goupil, Poulhalon and Schaeffer \cite{GPS}. 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 \cite{T,STW1}, which can also produce stable formulas for outer or inner plethysms, character values or branching multiplicities \cite{CT,STW2}. 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 \cite{FW} that the commutators of Goulden's operator with the operators $\alpha_{-k}=$ ``multiplication by $p_k$'' and their adjoints $\alpha_k$, 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 $\SG_n$ as the conjugacy class $\CC_{(2,1^{n-2})}$, one can view it as the sum $p_1(\Xi_n)=\sum_{i=1}^n\xi_i$ of the Jucys-Murphy elements, and look for a generating function of the differential operators $\DD_k$ implementing the multiplication by power sums $p_k(\Xi_n)$ for all $n$. One finds that the required generating function has a simple expression in terms of the classical vertex operator $\Gamma(z_1,z_2)$ describing the Fock space representations of $\gl_\infty$ and $\glchap_\infty$. Then, one can see that the brackets $[\DD_k,\alpha_l]$ generate a charge 1 representation of the Lie algebra ${\cal W}_{1+\infty}$, 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 \ref{sec:appl}, we remark that our vertex operator provides a new approach to the results of Kerov and Olshanski \cite{KO}. What we prove is that the coefficients of the products of power-sums of Jucys-Murphy elements $p_\mu(\Xi_n)$ on the normalized conjugacy classes of \cite{KO} are independent of $n$, which is equivalent to Proposition 3 of \cite{KO}. In Section \ref{sec:mat}, we give an alternative derivation of the operators of \cite{GPS} in terms of matrix integrals. We start, following Hanlon, Stanley and Stembridge \cite{HSS}, 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 $N\times N$ matrices, $N$ being sufficiently large (actually, it is the limit $N\rightarrow\infty$ 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 \cite{HSS}, whithout any reference to spherical functions (this answers a question raised in the last section of \cite{HSS}). Finally, we observe that the results of Kerov and Olshanski rederived in Section \ref{sec:appl} imply that the Goupil-Poulhalon-Schaeffer operators form a linear basis of the commutative subalgebra of $U({\cal W}_{1+\infty})$ generated by the $\DD_k$. \section{Notations and background}\label{sec:backg} \subsection{Symmetric functions} We denote by $Sym$ the (abstract) algebra of symmetric functions, with complex coefficients, and by $Sym(X)=Sym(x_1,\ldots,x_n)$ 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 $s_\lambda$ form an orthonormal basis. For $f\in Sym$, $D_f$ denotes the adjoint of the operator $g\mapsto fg$. For $A=\{a_1,a_2,\ldots\,\}$ we denote by $\sigma_z(A)=\prod_i(1-za_i)^{-1}$ and $\lambda_z(A)=\sigma_{-z}(A)^{-1}$ the generating series of complete and elementary symmetric functions of $A$. Other notations for symmetric functions are as in \cite{Mcd}. \subsection{The Frobenius characteristic map} The conjugacy class of permutations with cycle type $\mu$ is denoted by $\CC_\mu$. A central function $f$ on $\SG_n$ is identified to the element $F=\sum_\sigma f(\sigma)\sigma\in Z\SG_n$. The Frobenius characteristic map is the linear map $\ch: \C\SG_n\rightarrow Sym_n$ defined by $\ch(\sigma) =p_\mu$ is $\sigma$ is of cycle type $\mu$. The structure constants $c_{\alpha\beta}^\gamma$ of $Z\SG_n$ are defined by $\CC_\alpha \CC_\beta=\sum_\gamma c_{\alpha\beta}^\gamma \CC_\gamma$. The Frobenius map allows one to define a new product $\times$ on each $Sym_n$ by $\ch(F)\times\ch(G)=\ch(FG)$. We extend it to $Sym$ by setting $u\times v=0$ if $u$ and $v$ are homogeneous of different degrees. Then, $s_\lambda\times s_\mu = {1\over f^\lambda}\delta_{\lambda\mu}s_\lambda$, where $f^\lambda$ is the dimension of the representation $\lambda$ of $\SG_n$. We denote by $\Gamma$ the comultiplication dual to $\times$, that is, $\Gamma(s_\lambda) = {1\over f^\lambda}s_\lambda\otimes s_\lambda$. \subsection{Jucys-Murphy elements} The Jucys-Murphy elements of $\SG_n$ are the $n$ sums of transpositions \cite{Ju,Mu} \begin{equation} \xi_j=\xi_{j;n} =\sum_{ii_2>\ldots$, and $i_k=1-k$ for $k>>0$. Such a vector will be denoted by $|\lambda\>$, where the partition $\lambda$ is defined by $\lambda_k = i_k+k-1$. This space is the basic representation $L(\Lambda_0)$ of the affine Lie algebra $\glchap_\infty = {\mathfrak A}_\infty$, the universal central extension of the Lie algebra of $\Z\times \Z$-matrices with a finite number of nonzero diagonals. The generators $E_{ij}$ act in a simple way of the semi-infinite wedges. For $i\not=j$, 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 $i\le 0$, 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 $|\lambda\>\rightarrow s_\lambda$ (see \cite{Kac}, Chap. 14). More generally, the space ${\cal F}^{(m)}$ is spanned by wedges such that $i_k=1-k+m$ for $k>>0$. Their direct sum (for $m\in\Z$) is called the fermionic Fock space, and ${\cal F}^{(m)}$ is called the charge $m$ sector. \section{A vertex operator for power sums of Jucys-Murphy elements}\label{sec:vertex} \subsection{A differential operator for $\SG_n$} In this section, we will compute the differential operator $\DD^{(n)}$ acting on $Sym_n$ as the generating function \begin{equation} F_n(t) = \sum_{k\ge 1}p_k(\Xi_n)=\sum_{i= 1}^n(e^{t\xi_i}-1) \,, \end{equation} that is, for $P\in Sym_n$, $\DD^{(n)}P:=\ch(F_n(t))\times P$. We know that the eigenvalue of $p_k(\Xi_n)$ on the central idempotent $e_\lambda$ is $p_k(C(\lambda))$. Therefore, the eigenvalue of $\DD^{(n)}$ on $s_\lambda$ is \begin{equation} \sum_{\bo\in\lambda}\left(e^{tc_\bo}-1\right) =\sum_{\bo\in\lambda}(q^{c_\bo}-1) \end{equation} if we set $q=e^t$. This sum is easily evaluated in terms of the parts of $\lambda$: \begin{lemma} Let $\lambda$ be a non zero partition of length at most $n$. Then, $$ \sum_{\bo\in\lambda}q^{c_\bo}= {q\over q-1}\sum_{i=1}^n\left(q^{\lambda_i-i}-q^{-i}\right)\,. $$ \end{lemma} \Proof The contents of $\lambda$ are the numbers $-i+1,-i+2,\ldots,\lambda_i-1$ for $i=1,\ldots,\ell(\lambda)$. \qed Therefore, \begin{equation} \sum_{\bo\in\lambda}(q^{c_\bo}-1) ={q\over q-1}\sum_{i=1}^n\left(q^{\lambda_i-i}-q^{-i}\right) -\sum_{i=1}^n\lambda_i\,. \end{equation} Our first task is to express the operator induced by $\DD^{(n)}$ on the space of symmetric polynomials $Sym(x_1,\ldots,x_n)$ in terms of the variables $x_i$. We set \begin{equation} \Delta_n=\prod_{i$ for the scalar product of $Sym(A)$, we have $f(X)=\<\sigma_1(XA),f(A)\>$, so that $$ \DD^{(n)} f(X)=\<\DD^{(n)} \sigma_1(XA),f(A)\>=\<1,D_{\DD^{(n)} \sigma_1(XA)}f(A)\> =\<\sigma_1(XA),\DD f(A)\>\,. $$ Let $\nabla_i$ be the partial $q$-derivative with respect to $x_i$, {\it i.e.} $$ \nabla_i = {q^{D_i}-1\over (q-1)x_i}\,. $$ We have, for any function $f(X)=f(x_1,\ldots,x_n)$, $$ {q\over q-1}\left(\sum_{i=1}^n q^{\lambda_i-i}\right)f(X) =\left(\sum_{i=1}^n\nabla_i x_i\right)f(X) + \left(\sum_{i=1}^n [i]_{1/q}\right)f(X) \,. $$ To apply this to $f(X)=\Delta_n\sigma_1(XA)/\Box_n$, we note that \begin{equation} \left(\sum_{i=1}^n\nabla_i x_i\right) {\Delta_n\over\Box_n} = \left[ {n\over 1-q} + {q(1-q^{-n})\over (1-q)^2}\right] {\Delta_n\over\Box_n} \end{equation} and \begin{equation} \left(\sum_{i=1}^n\nabla_i x_i\right) {\Delta_n\over\Box_n}\sigma_1(XA) = \left(\sum_{i=1}^n\nabla_i x_i{\Delta_n\over\Box_n}\right)\sigma_1(XA) +\sum_{i=1}^n {qx_i\Delta^{(i)}\over q^n\Box_n} \nabla_i\sigma_1(XA) \end{equation} where $$ \Delta^{(i)}=q^{D_i}\Delta_n=\Delta_n A_i(X;q)\,, \quad A_i(X;q)=\prod_{j\not =i}{qx_i-x_j\over x_i-x_j}\,. $$ Hence, setting $\bar\DD^{(n)}=\DD^{(n)}+\EE^{(n)}$, where $\EE^{(n)}$ is the Euler operator, we have \begin{eqnarray*} \bar\DD^{(n)}\sigma_1(XA)&=& \left[\left({n\over 1-q} + {q(1-q^{-n})\over (1-q)^2}i\right) +\sum_{i=1}^n[i]_{1/q} \right]\sigma_1(XA)\\ &&+ q^{1-n}\sum_{i=1}^n x_i A_i(X;q)\nabla_i\sigma_1(XA) \\ &=&q^{-n}\sum_{i=1}^n A_i(X;q){qx_i\over(q-1)x_i} \left( {\sigma_{qx_i}(A)\sigma_1(XA)\over\sigma_{x_i}(A)}-\sigma_1(XA)\right) \\ &=&\sigma_1(XA) {q^{-n}\over q-1}\sum_{i=1}^n A_i(X;q)(\sigma_{qx_i}(A)\lambda_{-x_i}(A)-1) \\ &=&\sigma_1(XA) {q^{-n}\over q-1}\sum_{i=1}^n A_i(X;q)(\sigma_{x_i}((q-1)A)-1) \\ &=&\sigma_1(XA) {q^{-n}\over q-1}\sum_{m\ge 1}h_m((q-1)A)\sum_{i=1}^n A_i(X;q)x_i^m \\ &=&\sigma_1(XA) {q^{-n}\over q-1}\sum_{m\ge 1}h_m((q-1)A)q^n{h_m((1-q^{-1})A)\over 1-q^{-1}} \,. \end{eqnarray*} Rewriting this expression in a more symmetric form, we obtain the kernel of $\bar\DD=\DD+\EE$ \begin{equation} \bar K(X;A)={q\over (q-1)^2}\sum_{m\ge 1}q^{-m}h_m((q-1)A)h_m((q-1)X)\,. \end{equation} % The bosonization of the Euler operator being obviously $\EE=\sum_{k\ge 1} p_k D_{p_k}$, we have \begin{proposition} The differential operator corresponding to $\sum_{i\ge 1}(q^{\xi_i}-1)$ is $$ \DD={q\over (q-1)^2}\sum_{m\ge 1}q^{-m}h_m((q-1)A)D_{h_m((q-1)A)}-\EE \,. $$ \end{proposition} On this expression, it is clear that $\DD$ can be written $$\DD= {V_0-1\over (q-1)(1-q^{-1})}-\EE $$ where $V_0$ is the zero mode of the vertex operator \begin{eqnarray*} V(z;q)&=&\sigma_z((q-1)A)D_{\sigma_{1/z}((1-q^{-1})A)}\\ &=& \exp\left\{\sum_{k\ge 1}(q^k-1)p_k {z^k\over k}\right\} \exp\left\{\sum_{l\ge 1}(1-q^{-l})z^{-l}{\partial\over\partial p_l}\right\}\\ &=& :\exp\left\{\sum_{k\not = 0}{(1-q^{-k})z^{-k}\over k}\alpha_k\right\}:\\ &=&\sum_{m=-\infty}^\infty V_m(q)z^{-m}\,. \end{eqnarray*} % This operator satisfies the commutation relations \begin{equation} [V,\alpha_k]= z^k(1-q^k)V \quad (k\not = 0) \end{equation} so that \begin{equation} [V_l,\alpha_k]=(1-q^k)V_{k+l} \quad (k,l\in_Z,\ k\not = 0). \end{equation} In particular, \begin{equation}\label{eq:Vk} V_k = (1-q^k)^{-1}[V_0,\alpha_k] \quad (k\not = 0) \end{equation} that is, all the modes are generated by the action of the bosonic operators on $V_0$. \subsection{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 {\it e.g.}, \cite{Kac}, Corollary 14.10). In the notation of \cite{Kac}, $V(z;q)=\Gamma(qz,z)$, and if $\hat r_m$ denotes the representation of $\glchap_\infty$ in the charge $m$ sector ${\cal F}^{(m)}$ of the fermionic Fock space ${\cal F}$, one has in particular \begin{equation} W(z;q)={1\over 1-q^{-1}}\left(V(z;q)-1\right) =\sum_{i,j\in\Z}q^i z^j \hat r_0(E_{i,i-j})\,. \end{equation} Write $W(z;q)=\sum_k W_k(q) z^{-k}$, so that \begin{equation} W_k(q)=\hat r_0 \left(\sum_{i\in \Z} q^i E_{i,i+k}\right)\,. \end{equation} The commutation relations between the operators $W_k(q)$ are easily determined from the defining relations of $\glchap_\infty$, which read \begin{equation} [E_{ij},E_{kl}] = \delta_{jk}E_{il}-\delta_{li}E_{ki}+\Psi(E_{ij},E_{kl})c\, \end{equation} where $c$ is the central charge, and $\Psi$ is the 2-cocycle of $\gl_\infty$ given by \begin{equation} \Psi(E_{ij},E_{ji})=-\Psi(E_{ji},E_{ij})=1\ {\rm if}\ i\le 0,\ j\ge 1 \end{equation} and $\Psi(E_{ij},E_{kl})=0$ in all other cases. One has $\hat r_0(c)=1$, and a short calculation yields \begin{equation} [W_k(a),W_l(b)]=(b^k-a^l)W_{k+l}(ab) +\delta_{k,-l}{b^{-l}-a^{-k}\over 1-(ab)^{-1}}\,. \end{equation} One recognize that these relations are almost the standard presentation (in generating function form such as in \cite{KR}, Eq. (2.2.2)) of the Lie algebra usually denoted by ${\hat{\cal D}}$ or ${\cal W}_{1+\infty}$, the universal central extension of the Lie algebra ${\cal D}$ of all differential operators on the circle. The generators of ${\cal D}$ are the $z^kD^n$, where $D=z\partial_z$, and the corresponding elements of the central extension ${\hat{\cal D}}$ are denoted by $L_k^n$. The cocycle of the central extension is given by \begin{equation} \Phi(z^kf(D),z^lg(D))=\sum_{j\ge 1}kf(-j)g(k-j) \ {\rm if}\ k=-l\ge 0 \end{equation} and is 0 in all other cases. With this at hand, we see that the operators \begin{equation} T_k(q)=-q^{-1}W_k(q) \end{equation} satisfy \begin{equation} [T_k(a),T_l(b)] =(a^l-b^k)T_{k+l}(ab)+\delta_{k,-l} {a^{-k}-b^{-l}\over 1-ab} \end{equation} which is exactly Eq. (2.2.2) of \cite{KR} with $C=1$. Therefore, the coefficients $T_{k,n}$ defined by \begin{equation} T_k(e^t)=\sum_{n\ge 0} {t^n\over n!}T_{k,n} \end{equation} are the images of the $L^k_n$ in a representation $R_0$ of charge 1. Now, our differential operator $\DD$ reads \begin{eqnarray*} \DD &=&{-1 \over 1-q^{-1}} T_0(e^t)-\EE ={1\over t}{-t\over e^{-t}-1} \sum_{l\ge 0}{t^l\over l!}T_{0l}-\EE\\ &=& \sum_{n\ge 1}{t^{n}\over n!}\sum_{k=0}^{n}(-1)^{k-1} {n \choose k}B_k {T_{0,n+1-k}\over n+1-k}\\ &=& \sum_{n\ge 1}{t^{n}\over n!}\DD_n\,. \end{eqnarray*} We have $T_{00}=0$, $T_{01}=-\EE$, and Goulden's operator is $\DD_1=-{1\over 2}(T_{02}+T_{01})$. Also, $T_{k,1}=\alpha_k$, $T_{k,2}=2L_k$ where the $L_k$ are the charge 1 Virasoro operators considered in \cite{FW}. Since $[T_0(q),\alpha_k]=(1-q^k)T_k(q)$, we find that the result of \cite{FW} stating that the commutators $[\DD_1,\alpha_k]$ generate a Virasoro algebra can be extended as follows: \begin{proposition} The commutators $[\DD_j,\alpha_k]$ generate a charge 1 representation of ${\cal W}_{1+\infty}$.\qed \end{proposition} \subsection{Interpretation of the Virasoro operators} It would be of interest to have interpretations of the other generators $T_{ij}$ ($i\not =0$) in terms of natural operations on $Z\SG=\bigoplus_{n\ge 0}Z\SG_n$. As a step in this direction, we can propose such an interpretation for the positive part of the Virasoro algebra. Let, for $k\ge 1$ \begin{eqnarray} d'_k&=&\sum_{j\ge 0} p_j D_{p_{j+k}} =\sum_{j\ge 0}\alpha_{-j}\alpha_{j+k}\,,\\ d''_k &=& {1\over 2} \sum_{1\le i,j\atop i+j=k}\alpha_i\alpha_j\,,\\ d_k &=& d'_k+d''_k\,, \end{eqnarray} where $p_0=1$. Let also $\delta_k$, $\delta'_k$ and $\delta''_k$ be the linear maps $\C\SG_n\rightarrow\C\SG_{n-k}$ defined on permutations by \begin{eqnarray} \delta'_k(\sigma)&=&{1\over (l-1)(l-2)\cdots (l -k)}\sigma^{(k)}\,, \\ \delta''_k(\sigma)&=&{1\over 2 k!} \sum_{1\le i,j\atop i+j=k} \,i j\, \sigma^{(i,j)} \,,\\ \delta_k(\sigma)&=&\delta'_k(\sigma)+\delta''_k(\sigma)\,, \end{eqnarray} where $\sigma^{(k)}$ (resp. $\sigma^{(i,j)}$) are defined to be $0$ if $n,n-1,\ldots,n-k+1$ do not belong to the same cycle of length $l$ (resp. do not constitute two cycles of lengths $i$, $j$), and otherwise, $\sigma^{(k)}$ and $\sigma^{(i,j)}$ are the permutations whose cycle decomposition is obtained by erasing $n,n-1,\ldots,n-k+1$ in the cycle decomposition of $\sigma$. \begin{proposition} For $u\in Z\SG_n$, one has \begin{eqnarray*} {1\over (n)_k}\ch (\delta_k u) &=& d_k \ch(u)\,,\\ {1\over (n)_k}\ch (\delta'_k u) &=& d'_k \ch(u)\,. \end{eqnarray*} \end{proposition} \Proof A direct calculation. The numerical factors account for the orders of conjugacy classes, and could have been suppressed by using instead the basis $$ b_\mu={1\over n!}\sum_{\tau\in\SG_n} \tau^{-1}\sigma\tau $$ where $\sigma$ is any permutation of cycle type $\mu$. \qed The operators $d_i$ are the images of the generators $L_i$ of the Virasoro subalgebra of ${\cal W}_{1+\infty}$ under the above representation, while the $d'_i$ correspond to the Witt algebra. Therefore, $L_1$ corresponds to the map considered in \cite{KOV}, 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 $2$. \section{A stability property}\label{sec:appl} 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 $$ J_n(t)=\sum_{k\ge 0} \ch(p_k(\Xi_n)) {t^k\over k!}=\sum_{k\ge 0}J_n^k{t^k\over k!} $$ is equal to the constant term of ${1\over (q-1)(1-q^{-1})}(V(z;q)-1)p_1^n$, that is, to $$ {1\over (q-1)(1-q^{-1})}\left[ \sum_{k=0}^n{n\choose k} p_1^{n-k}h_k((q-1)A)(1-q^{-1})^k \right] $$ which has to be expanded with $q=e^t$ and $$ h_k((q-1)A)=\sum_{\mu\vdash k} \left[ \prod_i(e^{\mu_i t}-1)\right] z_\mu^{-1}p_\mu(A)\,. $$ This is best accomplished by means of a generating function. We have \begin{eqnarray*} {\cal J}(t) &=& \sum_{n\ge 0}{1\over n!}J_n(t) =[z^0]{(V(z;q)-1)e^{p_1}\over (q-1)(1-q^{-1})} \\ &=& e^{p_1}\sum_{k\ge 1}{(1-q^{-1})^{k-1}\over k!} {h_k((q-1)A\over q-1} \\ &=& e^{p_1}\sum_{k\ge 1}{(1-q^{-1})^{k-1}\over k!} \sum_{\kappa\vdash k}{p_\kappa(q-1)\over q-1}{p_\kappa(A)\over z_\kappa}\,. \end{eqnarray*} For $\kappa\vdash k\ge 1$, let \begin{equation} \phi_\kappa(t)={(1-q^{-1})^{k-1}\over k!z_\kappa}{p_\kappa(q-1)\over q-1}|_{q=e^t} \end{equation} so that if $\kappa=(1^{k_1}2^{k_2}\cdots\,)$, \begin{equation} \phi_\kappa(t)={t^{|\kappa|+\ell(\kappa)-2}\over k! k_1!k_2!\cdots} \left( 1+O(t)\right)\,, \end{equation} and \begin{eqnarray*} {\cal J}(t) &=&e^{p_1} \sum_{|\kappa|\ge 1}\phi_\kappa(t)p_\kappa(A)\\ &=& \sum_{n\ge 0}{1\over n!}\sum_{k=1}^n\phi_\kappa(t)\aa_{\kappa;n} \end{eqnarray*} where we have set $\aa_{\kappa;n}=(n)_kp_{\kappa,1^{n-k}}=\ch(a_{\kappa;n})$, where $a_{\kappa;n}$ are the normalized conjugacy classes defined in \cite{KO}. Hence, if \begin{equation} \phi_\kappa(t)=\sum_{m\ge |\kappa|+\ell(\kappa)-2}\phi_{\kappa;m} {t^m\over m!} \end{equation} we obtain \begin{equation} J_n^m=\ch( p_m(\Xi_n)) =\sum_{k=1}^{m+1} \sum_{\kappa\vdash k\atop \ell(\kappa)\le m-k+2} \phi_{\kappa;m} \aa_{\kappa;n}\,. \end{equation} Observe that the coefficients are independent of $n$. Actually, this is a special case of a result of Kerov and Olshanski \cite{KO}, which is equivalent to the existence of a similar $n$-independent expansion of all products of power sums $p_\mu(\Xi_n)$ as linear combinations of the $a_{\kappa;n}$. This more general result can also be obtained by the same method, but the expressions of the coefficients $\phi_{\kappa;\mu}$ become more cumbersome. Instead, we observe that if we can prove that the coefficients of the expansion $p_m((\Xi_n)\times \aa_{\kappa;n}$ on the basis $\aa_{\nu;n}$ are independent of $n$, the general result will follow by induction. To prove this, consider the generating function \begin{equation} G(t;A,B)=\sum_{n\ge 0}\sum_{m\ge 0} \sum_\kappa {1 \over n!m!} t^m J_n^m\times \aa_{\kappa;n}(A){p_\kappa(B)\over z_\kappa}\,. \end{equation} A calculation similar to the previous one (which is the case $B=0$) shows that \begin{equation} G(t;A,B)=e^{p_1(A)}\sigma_1(AB) \sum_{r\ge 1} {h_r((q-1)A)h_r((1-q^{-1})(B+E))\over (q-1)(1-q^{-1})} \end{equation} where symmetric functions of the ``exponential alphabet'' $E$ are defined by $\sigma_t(E)=e^t$ ({\it i.e.} $p_1(E)=1$ and $p_k(E)=0$ for $k>1$). On this expression, it is clear that the coefficient of ${t^m\over m!}{p_\kappa(B)\over z_\kappa}$ in $e^{-p_1(A)}G(t;A,B)$ is a polynomial $\sum_\mu d_{\kappa;m}^\mu p_\mu(A)$, so that \begin{equation} p_m(\Xi_n) a_{\kappa;n}=\sum_\mu d_{\kappa;m}^\mu a_{\mu;n} \end{equation} 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. \begin{eqnarray*} p_1(\Xi)&=&{1\over 2}a_2\\ p_2(\Xi)&=& {1\over 3} \,a_3+{1\over 2}\,a_{11}\\ p_3(\Xi)&=& {1\over 4} \,a_4+a_{21}+{1\over 2}\,a_2\\ p_4(\Xi)&=&{1\over 5} \,a_5+{1\over 2} \,a_{22}+a_{31}+{2\over 3}\,a_{111} +{5\over 3} \,a_3+{1\over 2}\,a_{11}\\ p_5(\Xi)&=&{1\over 6} \,a_6+a_{32}+a_{41}+{5\over 2} \,a_{211}+ {15\over 4}\,a_4+5 \,a_{21}+{1\over 2}\,a_2\\ p_6(\Xi)&=& {1\over 7} \,a_7+{1\over 2} \,a_{33}+a_{42}+a_{51}+3\,a_{221}+ 3 \,a_{311}+7 \,a_5\\ &&+{5\over 4} \,a_{1111}+{25\over 4} \,a_{22}+15 \,a_{31}+ {10\over 3} \,a_{111}+7 \,a_3+{1\over 2}\,a_{11}\\ \end{eqnarray*} One obtains the expression of each $p_m(\Xi_n)$ from the table by substituting $[(n-k)!]^{-1}z_{\kappa,1^{n-k}}\CC_{\kappa,1^{n-k}}$ to $a_\kappa$, $\kappa\vdash k$. For example, \begin{eqnarray*} p_2(\Xi_n)&=& {1\over 3}{3\cdot (n-3)!\over (n-3)!}\CC_{3,1^{n-3}} +{1\over 2} {n!\over (n-2)!}\CC_{1,1,1^{n-2}}\\ &=& \CC_{3,1^{n-3}}+{n\choose 2} \CC_{1^n}\,. \end{eqnarray*} \section{A matrix integral approach}\label{sec:mat} In this section, we will express generating functions for the coproducts of the elements $\aa_{\rho;n}$ as certain Gaussian integrals over the space of $N\times N$ complex matrices. Evaluating these integrals by Wick's formula, we obtain as a byproduct a new derivation of the differential operators of \cite{GPS}. The zonal spherical functions of the Gelfand pair $(GL(N,\C),U(N))$ are known to be expressible in terms of Schur functions (see \cite{Mcd}, Chap. VII, Sec. 5): $$ \Omega_\lambda(Z) ={s_\lambda(ZZ^*)\over s_\lambda(N)}\,. $$ As a consequence, we have closed form evaluations of the matrix integrals \begin{equation}\label{eq:matint} \int_{M_N(\C)} s_\lambda(AZBZ^*) d\nu(Z) =2^{|\lambda|}h(\lambda)s_\lambda(A)s_\lambda(B) \end{equation} where $h(\lambda)$ is the product of the hook-lengths of $\lambda$, $A$ and $B$ are arbitrary Hermitian matrices, and $d\nu$ is the Gaussian probability measure \begin{equation} d\nu(Z)=(2\pi)^{-N^2}e^{-{1\over 2}\tr(ZZ^*)}dZ\,,\quad dZ=\prod_{k,l=1}^N dx_{kl}dy_{kl}\,,\quad z_{kl}=x_{kl}+iy_{kl}\,. \end{equation} If $|\lambda|=n$, the right-hand side of (\ref{eq:matint}) is $$ 2^n n! \ {s_\lambda(A)s_\lambda(B)\over f_\lambda} =2^n n! \ \Gamma(s_\lambda)(A\otimes B) $$ where $\Gamma$ is the comultiplication dual to the $\times$-product, induced on $Sym$ by the convolution of central functions, and elements of $Sym\otimes Sym$ are interpreted as functions of tensor product of (square) matrices. Therefore, denoting by $u_\lambda^*$ the adjoint of a basis $u_\lambda$, \begin{eqnarray*} \Gamma(p_\lambda) &=&\sum_{\alpha,\beta} \<\Gamma(p_\lambda),p_\alpha^*\otimes p_\beta^*\>p_\alpha\otimes p_\beta\\ &=&{1\over (n!)^2}\sum_{\alpha,\beta} \p_\alpha\otimes p_\beta\\ &=&{1\over n!}\sum_{\alpha,\beta} c_{\alpha\beta}^\lambda p_\alpha\otimes p_\beta \end{eqnarray*} (where we have set $C_\alpha=\ch\CC_\alpha$), so that \cite{HSS} \begin{equation}\label{eq:HSS} \int_{M_N(\C)} p_\lambda(AZBZ^*)d\nu(Z) = 2^n\sum_{\alpha,\beta\vdash n} c_{\alpha\beta}^\lambda p_\alpha(A) p_\beta (B) \,. \end{equation} We will now form generating functions for the coproducts. Let $\rho$ be a partition of $r$. Then, \begin{eqnarray*} \int_{M_N(\C)}p_\rho(AZBZ^*){[p_1(AZBZ^*)]^{n-r}\over 2^{n-r}}d\nu(Z) &=&2^r\sum_{\alpha,\beta\vdash r}c^{\rho 1^{n-r}}_{\alpha\beta} p_\alpha(A)p_\beta(B)\\ & =& 2^rn!\ \Gamma(p_{\rho 1^{n-r}})\,. \end{eqnarray*} Therefore, the coproduct of the element $\aa_\rho=\sum_n \aa_{\rho;n}$ is given by \begin{eqnarray} \Gamma\left(\sum_{n\ge r}(n)_rp_{\rho 1^{n-r}}\right) &=&\int_{M_N(\C)}p_\rho(AZBZ^*)e^{{1\over 2}p_1(AZBZ^*)}d\nu(Z)\\ &=&\int_{M_N(\C)}p_\rho(AZBZ^*)d\mu(Z) \end{eqnarray} where \begin{equation} d\mu(Z)=d\mu_{A,B}(Z)=\pi^{N^2}e^{-{1\over 2}\tr(ZZ^*-AZBZ^*)}dZ \end{equation} 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, $\tr(ZZ^*-AZBZ^*)=\sum_{i,j}(1-a_ib_j)|z_{ij}|^2$. The total mass of $d\mu$ is \begin{equation} {\cal Z}=\int_{M_N(\C)}d\mu(Z)=\prod_{i,j}{1\over 1-a_ib_j}\,. \end{equation} % For a function $f$ on $M_N(\C)$, let \begin{equation} \={1\over {\cal Z}}\int_{M_N(\C)}f(Z)d\mu(Z) \end{equation} denote its expectation value. Since $d\mu$ is Gaussian, we can make use of Wick's formula, which in this contexts says the following: if $f_1,f_2,\ldots,f_m$ are $\R$-linear forms on $M_N(\C)$, we have \begin{eqnarray} \ &=& 0\, \\ \ &=& \Hf (\) \end{eqnarray} where the Hafnian of the matrix $(\)$ is defined by \begin{equation} \Hf (\)_{1\le i,j\le 2k} = \sum \prod_{i=1}^k \ \end{equation} the sum being taken over all pairs of $k$-uples $L=(l_1 &=&0\,, \\ \ &=&0\,, \\ \ &=& \delta_{il}\delta_{jk}{2\over 1-a_ib_j}\,. \end{eqnarray} Let $M=AZBZ^*$, and let $\sigma$ be the following permutation of cycle type $\rho$ \begin{equation}\label{eq:sigma} \sigma=(12\cdots\rho_1)(\rho_1+1,\rho_1+2,\cdots,\rho_1+\rho_2)\cdots(\cdots r)\,. \end{equation} Then, our generating function reads \begin{eqnarray*} \& =& \sum_{i_1,\ldots,i_r} M_{i_1,i_{\sigma(1)}}M_{i_2,i_{\sigma(2)}}\cdots M_{i_r,i_{\sigma(r)}}\\ &=&\sum_{i_1,\ldots,i_r\atop j_1,\ldots,j_r} a_{i_1}b_{j_1}a_{i_2}b_{j_2}\cdots a_{i_r}b_{j_r} \ \\ &=& \sum_{I,J}a_Ib_J\sum_{\tau\in\SG_r} \prod_{k=1}^r \delta_{i_k,i_{\sigma\tau(k)}}\delta_{j_k,j_{\tau(k)}} {2\over 1-a_{i_k}b_{j_k}}\\ &=& 2^r\sum_{I,J}\sum_{\tau\in\SG_r} G_{I,J} \prod_{k=1}^r \delta_{i_k,i_{\sigma\tau(k)}}\delta_{j_k,j_{\tau(k)}} \end{eqnarray*} where we have set $G_{i,j}=a_ib_j(1-a_ib_j)^{-1}$ and $G_{I,J}=\prod_k G_{i_k, j_k}$ Let us regard the multindices $I,J$ as functions $\{1,2,\ldots,r\}\rightarrow \N^*$. Then, the above product of Kronecker deltas is zero unless $I$ is constant on the orbits of $\sigma\tau$, and $J$ is constant on the orbits of $\tau$. To express the final result, we introduce the following notation. Given a permutation $\tau\in\SG_r$ and a vector $L=(l_1,l_2,\ldots,l_r)$ of positive integers, let \begin{equation} p_L^\tau = \prod_k\prod_{\gamma\in\, k- {\rm Cycles}(\tau)}p_{l_{\gamma_1}+l_{\gamma_2} +\cdots l_{\gamma_k}} \end{equation} product on all $k$-cycles $\gamma=(\gamma_1,\ldots,\gamma_k)$ of $\tau$. Then, \begin{proposition} As a symmetric function of the eigenvalues of $A$ and $B$, $$ \=2^r\sum_{l_1,\ldots,l_r\ge 1}\sum_{\tau\in\SG_r} p_L^{\sigma\tau}(A)p_L^{\tau}(B)\,. $$ where $\sigma$ is defined in (\ref{eq:sigma}). \end{proposition} \qed Note that the above calculation provides an answer to a question raised at the end of \cite{HSS}, namely, to find a direct combinatorial proof of (\ref{eq:HSS}). Indeed, to obtain the expectations with respect to $d\nu$ rather than $d\mu$, one just has to replace the propagators by $G_{i,j}=a_ib_j/2$, in which case the sum over $L$ disappears (the only remaining term being for $L=(1,1,\ldots,1)$) and one finds exactly (\ref{eq:HSS}). Now, if $\rho$ is a reduced partition (no part equal to 1), \begin{equation} \sigma_1(A\otimes B) \=2^r z_\rho\sum_{n\ge r}\Gamma(C_{\rho,1^{n-r}})(A\otimes B) \end{equation} and in general, the $\times$-multiplication by any symmetric function $F$ can be implemented by a differential operator as soon as its coproduct is known in the form \begin{equation} \Gamma(F)=\delta\sigma_1\sum f_{\alpha\beta}p_\alpha\otimes p_\beta\,, \end{equation} where $\delta$ is the comultiplication defined by $\delta(p_\mu)=p_\mu\otimes p_\mu$. Indeed, for any symmetric functions $G,H$, \begin{eqnarray*} \ &=& \ = \<\Gamma(F),H\otimes G\> \\ &=&\sum_{\alpha\beta}f_{\alpha\beta} \<\delta\sigma_1 p_\alpha\otimes p_\beta ,H\otimes G\>\\ &=&\sum_{\alpha\beta}f_{\alpha\beta} \<\delta\sigma_1 , D_{p_\alpha}H\otimes D_{p_\beta} G\> \\ &=& \sum_{\alpha\beta}f_{\alpha\beta} \<\sigma_1 , D_{p_\alpha}H\otimes D_{p_\beta} G\> \\ &=&\sum_{\alpha\beta}f_{\alpha\beta}\\\ &=& \left\<\sum_{\alpha\beta}f_{\alpha\beta}p_\alpha D_{p_\beta} G,H\right\>\,. \end{eqnarray*} Hence, we recover the result of \cite{GPS}, proving conjectures of Katriel \cite{Ka}: \begin{proposition} Let $\rho$ be a reduced partition of $r$. For any homogeneous symmetric function $G$ of degree $n\ge r$, $$ C_{\rho,1^{n-r}}\times G = \H_\rho (G) $$ where $\H_\rho$ is the differential operator $$ \H_\rho={1\over z_\rho} \sum_{l_1,\ldots,l_r\ge 1}\sum_{\tau\in\SG_r} p_L^{\sigma\tau}D_{p_L^{\tau}}\,. $$ \end{proposition} \qed \section{Final remarks}\label{sec:final} Recall that $p_\nu\times s_\lambda= (\chi^\lambda_\nu/ f^\lambda) s_\lambda$, where $f^\lambda$ is the number of standard tableaux of shape $\lambda$. Kerov and Olshanski have shown that for a partition $\rho$ of $r$, the function \begin{equation}\label{eq:KO} f_\rho(\lambda)= (n)_r{\chi^\lambda_{\rho,1^{n-r}}\over f^\lambda} \end{equation} is a polynomial in the ``shifted power-sums'' $$ \tilde p_k(\lambda)=\sum_{i\ge 1}(\lambda_i-i)^k-(-i)^k $$ which are the eigenvalues on $|\lambda\>=s_\lambda$ of the operators $$ P_k=\hat r_0\left(\sum_{i\in\Z}i^kE_{ii}\right) $$ of the Fock space representation of $\glchap_\infty$. (We have proved an equivalent result in Section \ref{sec:appl}.) A first consequence of this result is that the elements $\aa_{\rho;n}=\ch(a_{\rho;n})$ ($a_{\rho;n}\in Z\SG_n$ are the normalized conjugacy classes of \cite{KO}) have structure constants independent of $n$: there exist nonnegative integers $g_{\alpha\beta}^\gamma$ such that \begin{equation} \aa_{\alpha;n}\times \aa_{\beta;n} = \sum_\gamma g_{\alpha\beta}^\gamma \aa_{\gamma;n} \end{equation} for all $n$. Therefore, the operators $\A_\rho$ implementing simultaneously the multiplication by all $\aa_{\rho;n}$ also satisfy \begin{equation} \A_{\alpha}\A_{\beta} = \sum_\gamma g_{\alpha\beta}^\gamma \A_{\gamma} \end{equation} so that they form a linear basis of commutative subalgebra of $U(\glchap_\infty)$. When $\rho$ is reduced, $\A_{\rho}=z_\rho\H_\rho$. A second consequence is that these operators actually belong to the image of $U({\cal W}_{1+\infty})$ under the representation $R_0$ of Section \ref{sec:vertex}. Indeed, (\ref{eq:KO}) show that $\A_\rho$ is a polynomial in the commuting operators $P_k$, which are related to the $\DD_k$ of the previous section by \begin{equation} P_n=\sum_{k=0}^{n-1}{n\choose k}\DD_k \end{equation} where we have set $\DD_0=\EE$. Kerov and Olshanski also identified the algebra of the $a_{\rho;n}$ to an algebra of differential operators $A_{\rho;N}$ living in the center of $U(\gl_N)$ for $N\ge n$. 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 $$ \tilde p_k(D)=\sum_{i=1}^N D_i^k-(-i)^k $$ and $f_\rho(\lambda)=\sum_\nu k_{\rho\nu}\tilde p_\nu(\lambda)$, the restriction of $A_{\rho;N}$ to invariant functions is given by \begin{equation} A_{\rho;N} = {\Box_N\over \Delta_N} \sum_\nu k_{\rho\nu}\tilde p_\nu(D) {\Delta_N\over \Box_N}\,. \end{equation} \begin{thebibliography}{abc} % \bibitem{AKOS}{H. Awata, H. Kubo, S. Odake, and J. Shiraishi}, {\it Quantum ${W}_N$ algebras and Macdonald polynomials}, Comm. Math. Phys. {\bf 179} (1996), 401--416. % \bibitem{CT}{C. Carr\'e and J.-Y. Thibon}, {Plethysm and vertex operators}, Adv. Appl. Math. {\bf 13} (1992), 390-403. % \bibitem{FW}{I.B. Frenkel and W. Wang}, {Virasoro algebra and wreath product convolution}, preprint {\tt QA/0006087}. % \bibitem{Go}{I. Goulden}, {\it A differential operator for symmetric functions and the combinatorics of multiplying transpositions}, Trans. Amer. Math. Soc. {\bf 344} (1994), 421--440. % \bibitem{GPS}{A. Goupil, D. Poulalhon and G. Schaeffer}, {\it Central characters and conjugacy classes of the symmetric group}, Proceedings of FPSAC'00 (D. Krob, A.A. Mikhalev and A.V. Mikhalev eds.), Moscow, June 2000 (Springer), 238--249. % \bibitem{HSS}{P.J. Hanlon, R.P. Stanley and J.R. Stembridge}, {\it Some combinatorial aspects of spectra of normally distributed random matrices}, Contemp. Math. {\bf 138} (1992), 151--174. % \bibitem{Ju}{A. Jucys}, {\it Symmetric polynomials and the center of the symmetric group rings}, Rep. Math. Phys. {\bf 5} (1974), 107--112. % \bibitem{Kac}{V.G. Kac}, {\it Infinite dimensional Lie algebras}, 3rd edition, Cambridge, 1990. \bibitem{Ka}{J. Katriel},{\it The class algebra of the symmetric group}, Proc. 10th FPSAC Conference, N. Bergeron and F. Sottile Eds., Toronto (1998),401-410. % \bibitem{KR}{V.G. Kac and A. Radul}, {\it Quasifinite highest weight modules over the Lie algebra of differential operators on the circle}, Comm. Math. Phys. {\bf 157} (1993), 429--457. % \bibitem{KO}{S. Kerov and G. Olshanski}, {\it Polynomial functions on the set of Young diagrams}, C.R. Acad. Sci. Paris S\'er. I {\bf 319} (1994), 121--126. % \bibitem{KOV}{S. Kerov, G. Olshansi and A. Vershik}, {\it Harmonic analysis on the infinite symmetric group. A deformation of the regular representation}, C.R. Acad. Sci. Paris S\'er. I {\bf 316} (1993), 773--778. % \bibitem{Mcd}{I.G. Macdonald}, {\it Symmetric functions and Hall polynomials}, 2nd edition, Oxford, 1995. % \bibitem{Mu}{G. Murphy}, {\it A new construction of Young's seminormal representation of the symmetric group}, J. Algebra {\bf 69} (1981), 287--291. % \bibitem{OO}{A. Okounkov and G. Olshanski}, {\it Shifted Schur functions}, St. Petersburg Math. J. {\bf 9} (1998), 239--300. \bibitem{STW1}{T. Scharf, J.-Y. Thibon and B.G. Wybourne} {\it Reduced notation, inner plethysm and the symmetric group}, J. Phys. A {\bf 24} (1993), 7461--7478. % \bibitem{STW2}{T. Scharf, J.-Y. Thibon and B.G. Wybourne} {\it Generating functions for stable branching coefficients of $U(n)\downarrow S_n$, $O(n)\downarrow S_n$ and $O(n)\downarrow S_{n-1}$}, J. Phys. A {\bf 30} (1997), 6963--6975. % \bibitem{T}{J.-Y. Thibon}, {\it Hopf algebras of symmetric functions and tensor products of symmetric group representations}, Internat. J. Alg. Comp. {\bf 2} (1991), 207--221. % \end{thebibliography} \bigskip {\footnotesize\noindent \sc Institut Gaspard Monge, \\ Universit\'e de Marne-la-Vall\'ee,\\ 77454 Marne-la-Vall\'ee cedex,\\ FRANCE} \end{document}