Jantzen filtration of Weyl modules for general linear supergroups

TL;DR

This paper constructs the Jantzen filtration for Weyl modules of the general linear supergroup over fields of odd characteristic, providing a sum formula for their characters and characterizing when irreducible modules are Kac modules.

Contribution

It introduces a method to construct the Jantzen filtration for Weyl modules of $GL(m|n)$ with typical weights and derives a character sum formula, linking irreducibility to $p$-typicality.

Findings

Jantzen filtration constructed for Weyl modules with typical weights

Sum formula for characters of these modules derived

Irreducible modules are Kac modules iff they are $p$-typical

Abstract

Let $G=GL(m|n)$ be a general linear supergroup over an algebraically closed field $k$ of odd characteristic $p$. In this paper we construct Jantzen filtration of Weyl modules $V(\lambda)$ of $G$ when $\lambda$ is a typical weight in the sense of Kac's definition, and consequently obtain a sum formula for their characters. By Steinberg's tensor product theorem, it is enough for us to study typical weights with aim to formulate irreducible characters. As an application, it turns out that an irreducible $G$-module $L(\lambda)$ can be realized as a Kac module if and only if $\lambda$ is $p$-typical.