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

About this Title

Alexander Kleshchev, Department of Mathematics, University of Oregon, Eugene, Oregon 97403 and Vladimir Shchigolev, Department of Algebra, Faculty of Mathematics, Lomonosov Moscow State University, Leninskiye Gory, Moscow 119899, Russia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 220, Number 1034
ISBNs: 978-0-8218-7431-8 (print); 978-0-8218-9205-3 (online)
Published electronically: March 7, 2012
Keywords: projective representations, symmetric groups, Lie superalgebras
MSC: Primary 20C30; Secondary 20C25, 20C20, 17B10.

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Table of Contents


  • Introduction
  • 1. Preliminaries
  • 2. Lowering operators


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.