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dc.contributor.authorBiswas, Shibananda
dc.contributor.authorGhosh, Gargi
dc.contributor.authorMisra, Gadadhar
dc.contributor.authorRoy, Subrata Shyam
dc.date.accessioned2017-09-07T09:49:56Z
dc.date.available2017-09-07T09:49:56Z
dc.date.issued2017-07-20
dc.identifier.urihttps://arxiv.org/abs/1707.02956
dc.identifier.urihttp://publications.mfo.de/handle/mfo/1308
dc.descriptionMSC: 47A13; 47B32; 20B30en_US
dc.descriptionOWLF 2017en
dc.description.abstractThe 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.en_US
dc.language.isoenen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2017,19
dc.subjectHilbert Modulesen_US
dc.subjectSymmetric Functionsen_US
dc.titleReducing sub-modules of the Bergman module $\mathbb A^{(\lambda)}(\mathbb D^n)$ under the action of the symmetric groupen_US
dc.typePreprinten_US
dc.rights.licenseDieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.de
dc.rights.licenseThis document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.en
dc.identifier.doi10.14760/OWP-2017-19
local.scientificprogramOWLF 2017en_US
local.series.idOWP-2017-19
local.subject.msc47
local.subject.msc20
dc.identifier.urnurn:nbn:de:101:1-2021032310103683417519
dc.identifier.ppn1655550640


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