# Mathematical Sciences Research Institute

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# Deligne categories and complexes of representations of symmetric groups

## Connections for Women: Group Representation Theory and Applications February 01, 2018 - February 02, 2018

February 02, 2018 (02:15 PM PST - 03:00 PM PST)
Speaker(s): Inna Entova-Aizenbud (Ben Gurion University of the Negev)
Location: MSRI:
Tags/Keywords
• Deligne categories

• infinite symmetric groups

• representations of symmetric groups

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

#### 7-Entova-Aizenbud

Abstract

Let $V$ be a finite-dimensional (complex) vector space, and $Sym(V)$ be the symmetric algebra on this vector space. We can consider the multiplication map $Sym(V) \otimes V \to V$ as a complex of $GL(V)$-representations of length $2$. I this talk, I will describe how tensor powers of the above complex define interesting complexes of representations of the symmetric group $S_n$, which were studied by Deligne in the paper "La Categorie des Representations du Groupe Symetrique $S_t$, lorsque $t$ n’est pas un Entier Naturel". I will then explain how computing the cohomology of these complexes helps establish a relation between the Deligne categories and the representations of $S_{\infty}$, which are two natural settings for studying stabilization in the theory of finite-dimensional representations of the symmetric groups. This is joint work with D. Barter and Th. Heidersdorf

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