English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The weighted Bergman spaces on the polydisc, $\mathbb A^{(\lambda)}(\mathbbD^n)$, $\lambda>0,$ splits into orthogonal direct sum of subspaces $\mathbbP_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by thepartitions $\boldsymbol p$ of $n,$ which are in one to one correspondence withthe equivalence classes of the irreducible representations of the symmetricgroup on $n$ symbols. In this paper, we prove that each sub-module $\mathbbP_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locallyfree Hilbert module of rank equal to square of the dimension $\chi_{\boldsymbolp}(1)$ of the corresponding irreducible representation. It is shown that giventwo partitions $\boldsymbol p$ and $\boldsymbol q$, if $\chi_{\boldsymbol p}(1)\ne \chi_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbolp}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbolq}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ are not equivalent. We provethat for the trivial and the sign representation corresponding to thepartitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$,respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbbA^{(\lambda)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbbA^{(\lambda)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, weshow that all the sub-modules in this decomposition are inequivalent.

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