On the Brauer-Picard group of a finite symmetric tensor category

TL;DR

This paper computes the Brauer-Picard group of a specific finite symmetric tensor category related to supergroups, revealing its structure via symplectic group actions on a Lagrangian Grassmannian.

Contribution

It explicitly determines the Brauer-Picard group of the category C_n by linking it to the action of a projective symplectic group on a classical geometric object.

Findings

BrPic(C_n) is identified with a group of braided tensor autoequivalences.

The action of this group corresponds to a projective symplectic group action.

The study connects categorical autoequivalences with classical symplectic geometry.

Abstract

Let C_n denote the representation category of a finite supergroup generated by purely odd n-dimensional vector space. We compute the Brauer-Picard group BrPic(C_n) of C_n. This is done by identifying BrPic(C_n) with the group of braided tensor autoequivalences of the Drinfeld center of C_n and studying the action of the latter group on the categorical Lagrangian Grassmannian of C_n. We show that this action corresponds to the action of a projective symplectic group on a classical Lagrangian Grassmannian.