Reducing sub-modules of the Bergman module A(λ)(Dn) under the action of the symmetric group

Abstract

The weighted Bergman spaces on the polydisc, A(λ)(Dn), \lambda>0, splits into orthogonal direct sum of subspaces Pp(A(λ)(Dn)) indexed by the partitions p of n, which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on n symbols. In this paper, we prove that each sub-module Pp(A(λ)(Dn)) is a locally free Hilbert module of rank equal to square of the dimension χp(1) of the corresponding irreducible representation. It is shown that given two partitions p and q, if χp(1)≠χq(1), then the sub-modules Pp(A(λ)(Dn)) and Pq(A(λ)(Dn)) are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions p=(n) and p=(1,…,1), respectively, the sub-modules P(n)(A(λ)(Dn)) and P(1,…,1)(A(λ)Dn)) are inequivalent. In particular, for n=3, we show that all the sub-modules in this decomposition are inequivalent.