$GL(m|n)$-supermodules with good and Weyl filtrations

TL;DR

This paper establishes criteria for $GL(m|n)$-supermodules to possess good or Weyl filtrations, introduces Steinberg supermodules, and identifies new finite-dimensional tilting supermodules.

Contribution

It provides necessary and sufficient conditions for filtrations, defines Steinberg supermodules, and discovers new tilting supermodules in the superalgebra context.

Findings

Criteria for good and Weyl filtrations established

Steinberg supermodules introduced and their properties analyzed

New finite-dimensional tilting supermodules identified

Abstract

The purpose of this paper is to prove necessary and sufficient criteria for a $GL(m|n)$-supermodule to have a good or Weryl filtration. We also introduce the notion of a Steinberg supermodule analogous to the classical notion of Steinberg module. We prove that the Steinberg supermodule inherits some properties of the Steinberg module. Some new series of finite-dimensional tilting supermodules are found.