Deligne categories and representations of the infinite symmetric group

Daniel Barter, Inna Entova-Aizenbud, Thorsten Heidersdorf

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We establish a connection between two settings of representation stability for the symmetric groups Sn over C. One is the symmetric monoidal category Rep(S) of algebraic representations of the infinite symmetric group S=⋃nSn, related to the theory of FI-modules. The other is the family of rigid symmetric monoidal Deligne categories Rep_(St), t∈C, together with their abelian versions Rep_ab(St), constructed by Comes and Ostrik. We show that for any t∈C the natural functor Rep(S)→Rep_ab(St) is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of S. Considering the highest weight structure on Rep_ab(St), we show that the image of any object of Rep(S) has a filtration with standard objects in Rep_ab(St). As a by-product of the proof, we give answers to the questions posed by P. Deligne concerning the cohomology of some complexes in the Deligne category Rep_(St), and their specializations at non-negative integers n.

Original languageEnglish
Pages (from-to)1-47
Number of pages47
JournalAdvances in Mathematics
Volume346
DOIs
StatePublished - 13 Apr 2019

Keywords

  • Deligne categories
  • FI-modules
  • Representations of the infinite symmetric group
  • Stable representation theory
  • Symmetric group
  • Tensor categories

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