Reducing sub-modules of the Bergman module ${\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n)$ under the action of the symmetric group

Abstract

The weighted Bergman spaces on the polydisc, ${\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n)$, $\lambda >0,$ splits into orthogonal direct sum of subspaces ${\mathbb{P}}_{\mathbf{p}}({\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n))$ indexed by the partitions $\mathbf{p}$ 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 ${\mathbb{P}}_{\mathbf{p}}({\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n))$ is a locally free Hilbert module of rank equal to square of the dimension ${\chi }_{\mathbf{p}}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\mathbf{p}$ and $\mathbf{q}$, if ${\chi }_{\mathbf{p}}(1)\ne {\chi }_{\mathbf{q}}(1),$ then the sub-modules ${\mathbb{P}}_{\mathbf{p}}({\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n))$ and ${\mathbb{P}}_{\mathbf{q}}({\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n))$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\mathbf{p}=(n)$ and $\mathbf{p}=(1,\dots ,1)$, respectively, the sub-modules ${\mathbb{P}}_{(n)}({\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n))$ and ${\mathbb{P}}_{(1,\dots ,1)}({\mathbb{A}}^{(\lambda )}({\mathbb{D}}^n))$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.

22 pages