Cohomological Tensor Functors on Representations of the General Linear Supergroup

TL;DR

This paper introduces cohomological tensor functors for representations of the supergroup $Gl(n|n)$, providing explicit formulas for their images and applications to spectral sequences and superdimensions.

Contribution

It defines and analyzes cohomological tensor functors on supergroup representations, including a formula for the image of irreducible representations under these functors.

Findings

The functor DS is semisimple and multiplicity free on irreducible representations.

Derived formulas for the superdimension of irreducible representations.

Applications include degeneration of spectral sequences.

Abstract

We define and study cohomological tensor functors from the category $T_n$ of finite-dimensional representations of the supergroup $Gl(n|n)$ into $T_{n-r}$ for $0 <r \leq n$. In the case $DS: T_n \to T_{n-1}$ we prove a formula $DS(L) = \bigoplus \Pi^{n_i} L_i$ for the image of an arbitrary irreducible representation. In particular $DS(L)$ is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation.