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 $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ 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 $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $\chi_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $\chi_{\boldsymbol p}(1) \ne \chi_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol p}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.