Branching graphs for finite unitary groups in non-defining characteristic

TL;DR

This paper links the modular branching rule for unipotent modules of finite unitary groups to crystal graphs of Fock spaces, providing combinatorial formulas and confirming a recent conjecture.

Contribution

It introduces a piecewise description of the modular branching rule using crystal graphs and offers a combinatorial formula for modules from cuspidal modules of defect 0.

Findings

Branching rules are described by connected components of crystal graphs.

A combinatorial formula connects modules from cuspidal modules of defect 0.

Partially proves a conjecture of Jacon and collaborators.

Abstract

We show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favourable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.