On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$

TL;DR

This paper investigates the structure of tensor categories related to irreducible representations of the supergroup $GL(n|n)$, revealing their semisimple quotients and subgroup structures, and analyzing tensor product decompositions.

Contribution

It determines the structure of the semisimple quotient categories and the associated algebraic groups for $GL(n|n)$ representations, providing new insights into their tensor product decompositions.

Findings

Identification of the semisimple tannakian category $Rep(H_n)$ as a quotient of $ ext{Rep}(GL(n|n))$

Determination of the connected derived subgroup $G_n$ and related groups $G_ u$

Structural description of tensor product decomposition laws up to superdimension zero

Abstract

We study the quotient of $\mathcal{T}_n = Rep(GL(n|n))$ by the tensor ideal of negligible morphisms. If we consider the full subcategory $\mathcal{T}_n^+$ of $\mathcal{T}_n$ of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $Rep(H_n)$ where $H_n$ is a pro-reductive algebraic group. We determine the connected derived subgroup $G_n \subset H_n$ and the groups $G_{\lambda} = (H_{\lambda}^0)_{der}$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(\lambda)$. This gives structural information about the tensor category $Rep(GL(n|n))$, including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on $2$-torsion in $\pi_0(H_n)$.