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Irreducible restrictions of spin representations of symmetric and alternating groups

Alexander Kleshchev, Lucia Morotti, Pham Huu Tiep

TL;DR

This paper addresses irreducible restrictions of spin representations from the Schur double covers ${\hat{\mathsf S}}_n$ and ${\hat{\mathsf A}}_n$ in odd characteristic $p>2$, focusing on imprimitive subgroups and leveraging a blend of modular spin representation theory and partition combinatorics. It develops a unified framework using twisted group algebras, Sergeev-type algebras, and $p$-strict partitions to classify irreducible restrictions, and provides deep reduction lemmas, branching rules, and regularization techniques that culminate in a strengthened imprimitive restriction theorem. Key contributions include explicit invariant calculations under wreath-product subgroups, a detailed reduction-mod-$p$ theory for basic and second-basic spin modules, and practical criteria for reducibility via special homomorphisms. The results enhance the understanding of maximal subgroup structures in finite classical groups by clarifying when spin representations retain irreducibility upon restriction and by mapping out the intricate modular behavior of spin modules under various subgroups.

Abstract

Let $\mathbb{F}$ be an algebraically closed field and $G$ be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups $H<G$ and $\mathbb{F} G$-modules $L$ such that the restriction $L\downarrow_H$ is irreducible. This problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $G$ is the Schur's double cover of alternating or symmetric group.

Irreducible restrictions of spin representations of symmetric and alternating groups

TL;DR

This paper addresses irreducible restrictions of spin representations from the Schur double covers and in odd characteristic , focusing on imprimitive subgroups and leveraging a blend of modular spin representation theory and partition combinatorics. It develops a unified framework using twisted group algebras, Sergeev-type algebras, and -strict partitions to classify irreducible restrictions, and provides deep reduction lemmas, branching rules, and regularization techniques that culminate in a strengthened imprimitive restriction theorem. Key contributions include explicit invariant calculations under wreath-product subgroups, a detailed reduction-mod- theory for basic and second-basic spin modules, and practical criteria for reducibility via special homomorphisms. The results enhance the understanding of maximal subgroup structures in finite classical groups by clarifying when spin representations retain irreducibility upon restriction and by mapping out the intricate modular behavior of spin modules under various subgroups.

Abstract

Let be an algebraically closed field and be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups and -modules such that the restriction is irreducible. This problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where is the Schur's double cover of alternating or symmetric group.

Paper Structure

This paper contains 49 sections, 123 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Ground field
  4. Groups and modules
  5. Superalgebras and supermodules
  6. Combinatorics of partitions
  7. Compositions, partitions and tableaux
  8. $p$-strict and $p$-restricted partitions
  9. Addable and removable nodes for $p$-strict partitions
  10. Representations of symmetric groups
  11. Symmetric and alternating groups
  12. Modules over symmetric groups
  13. Two-row partitions
  14. Invariants
  15. Invariants $(S_k^*)^{\mathsf W}$ for $k\leq p$
  16. ...and 34 more sections

Key Result

Theorem A

KT Let $G = {\hat{\mathsf S}}_n$ or ${\hat{\mathsf A}}_n$ with $n \geq 5$, $L$ be a faithful irreducible ${\mathbb F} G$-module, and let $H$ be a subgroup of $G$ such that $\pi(H)<{\mathsf S}_n$ is a primitive subgroup not containing ${\mathsf A}_n$. Then $L{{\downarrow}}_{H}$ is irreducible if and

Theorems & Definitions (218)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 208 more