Mixed Tensors of the General Linear Supergroup

TL;DR

This paper explores the structure of mixed tensors in the general linear supergroup, providing explicit decompositions, character formulas, and classifications for various cases, advancing understanding of supergroup representation theory.

Contribution

It introduces a detailed analysis of mixed tensors in $GL(m|n)$, including explicit tensor product decompositions and classifications for superdimension and atypicality.

Findings

Explicit tensor product decompositions for projective and Kostant modules

Character and dimension formulas for $GL(m|n)$

Classification of mixed tensors with non-zero superdimension

Abstract

We describe the image of the canonical tensor functor from Deligne's interpolating category $Rep(GL_{m-n})$ to $Rep(GL(m|n))$ attached to the standard representation. This implies explicit tensor product decompositions between any two projective modules and any two Kostant modules of $GL(m|n)$, covering the decomposition between any two irreducible $GL(m|1)$-representations. We also obtain character and dimension formulas. For $m>n$ we classify the mixed tensors with non-vanishing superdimension. For $m=n$ we characterize the maximally atypical mixed tensors and show some applications regarding tensor products.