Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup Q(n)

TL;DR

This paper develops a new theory of lowering operators for the supergroup Q(n), connecting two approaches to projective representation theory of symmetric groups and deriving new branching rules.

Contribution

It introduces a novel theory of lowering operators for Q(n) that links Sergeev duality with crystal graph methods, advancing the understanding of projective representations.

Findings

Developed the theory of lowering operators for Q(n)

Connected two major approaches in projective representation theory

Derived new branching rules for symmetric group representations

Abstract

There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups. This is achieved by developing the theory of lowering operators for the supergroup $Q(n)$ which is parallel to (although much more intricate than) the similar theory for $GL(n)$ developed by the first author. The theory of lowering operators for $GL(n)$ is a non-trivial generalization of Carter's work in characteristic zero, and it has received a lot of attention. So this part of our work might be of independent interest. One of the applications of lowering operators is to tensor products of irreducible $Q(n)$-modules with natural and dual natural modules, which leads to important special translation functors. We describe the socles and primitive vectors in such tensor products.