Semisimplifications and representations of the General Linear Supergroup

TL;DR

This paper investigates the semisimplification of the category generated by finite-dimensional irreducible representations of the algebraic supergroup $GL(m|n)$, revealing a canonical decomposition that simplifies understanding its structure.

Contribution

It provides a canonical decomposition of the pro-reductive group associated with $GL(m|n)$, reducing the problem to the equal rank case $m=n$.

Findings

Semisimplification equivalent to representations of a pro-reductive group $H_{m|n}$

Canonical decomposition $H_{m|n} o GL(m-n) imes H_{n|n}$

Reduction of the problem to the equal rank case $m=n$

Abstract

We study the semisimplification of the full karoubian subcategory generated by the irreducible finite dimensional representations of the algebraic supergroup $GL(m|n)$ over an algebraically closed field of characteristic zero. This semisimplification is equivalent to the representations of a pro-reductive group $H_{m|n}$. We show that there is a canonical decomposition $H_{m|n} \cong GL(m\!-\! n) \times H_{n|n}$, thereby reducing the determination of $H_{m|n}$ to the equal rank case $m\! =\! n$ which was treated in a previous paper.