Modular representations in type A with a two-row nilpotent central character

TL;DR

This paper explores the representation theory of rak{sl}_{m+2n} in positive characteristic with a focus on two-row nilpotent p-characters, providing combinatorial dimension formulas and analyzing simple objects via geometric and categorification techniques.

Contribution

It introduces combinatorial dimension formulas for simple modules and studies their structure using geometric categorification, extending previous geometric parametrizations.

Findings

Derived explicit combinatorial dimension formulas for simple modules.

Computed Jordan-Holder multiplicities for baby Vermas in specific cases.

Linked the results to positive characteristic analogues of classical character formulas.

Abstract

We study the category of representations of $\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Holder multiplicities of the simples inside the baby Vermas (in special case where n=1, i.e. that a subregular nilpotent, these were known from work of Jantzen). We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the [BMR] equivalence. The dimension formulae may be viewed as a positive characteristic analogue of the combinatorial character formulae for simple objects in parabolic category O for $\mathfrak{sl}_{m+2n}$, due to Lascoux and Schutzenberger.