A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroups

TL;DR

This paper develops a combinatorial framework for understanding Donkin-Koppinen filtrations of the coordinate algebra of general linear supergroups, utilizing superderivations and bideterminants to explicitly describe module structures.

Contribution

It introduces a specific ordering of weights and applies combinatorial techniques to explicitly construct the filtration and basis of modules for general linear supergroups.

Findings

Computed the action of odd superderivations on generators.

Established a weight ordering for filtrations.

Determined a basis of modules in the filtration.

Abstract

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $\phi^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $\Gamma$ be a finitely generated ideal $\Gamma$ of $X(T)^+$ and $O_{\Gamma}(K[G])$ be the largest $\Gamma$-subsupermodule of $K[G]$ having simple composition factors of highest weights $\lambda\in \Gamma$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_{\Gamma}(K[G])$.