Combinatorial aspects of the linkage principle for general linear supergroups

TL;DR

This paper investigates the combinatorial structure of the linkage principle for general linear supergroups, focusing on primitive vectors, supermodules, and morphisms to understand odd linkage in characteristic not equal to 2.

Contribution

It provides explicit descriptions of primitive vectors and morphisms to analyze odd linkage for general linear supergroups, extending the understanding of their representation theory.

Findings

Explicit description of $G_{ev}$-primitive vectors in supermodules

Results on properties of morphisms $\psi_k$ related to odd linkage

Insights into the structure of supermodules over fields with characteristic not 2

Abstract

Let $G=GL(m|n)$ be a general linear supergroup and $G_{ev}$ be its even subsupergroup isomorphic to $GL(m)\times GL(n)$. In this paper we use the explicit description of $G_{ev}$-primitive vectors in the costandard supermodule $\nabla(\lambda)$, the largest polynomial $G$-subsupermodule of the induced supermodule $H^0_G(\lambda)$, for $(m|n)$-hook partition $\lambda$, and a properties of certain morphisms $\psi_k$ to derive results related to the odd linkage for $G$ over a field $F$ of characteristic different from $2$.