Crystal structures arising from representations of $GL(m|n)$

TL;DR

This paper explores the modular representation theory of the supergroup $GL(m|n)$, revealing how crystal graph combinatorics describe its category $ ext{O}$ and related structures over arbitrary characteristic fields.

Contribution

It introduces a combinatorial framework using crystal graphs to understand the representation theory of $GL(m|n)$ in modular settings, including linkage principles and translation functors.

Findings

Crystal graphs describe the modular category $ ext{O}$ for $GL(m|n)$

Linkage principles are established for supermodules

Serganova's odd reflections induce canonical crystal isomorphisms

Abstract

This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that $GL(m|n)$ has non-conjugate Borel subgroups and we show how Serganova's odd reflections give rise to canonical crystal isomorphisms.