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Irreducible projective representations of the symmetric group which remain irreducible in characteristic $2$

Matthew Fayers

Abstract

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan--Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.

Irreducible projective representations of the symmetric group which remain irreducible in characteristic $2$

Abstract

For any finite group and any prime one can ask which ordinary irreducible representations remain irreducible in characteristic . We answer this question for when is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan--Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.

Paper Structure

This paper contains 28 sections, 18 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Symmetric groups, Hecke algebras and Schur algebras
  4. Double covers of $\sss n$ and projective representations
  5. Partitions
  6. $2$-cores and $2$-quotients
  7. Irreducible modules, irreducible characters and decomposition numbers
  8. The inverse of the decomposition matrix of the Schur algebra
  9. The degree of an irreducible spin character
  10. Regularisation and doubling
  11. Blocks
  12. Branching rules
  13. Symmetric functions
  14. $2$-Carter partitions and the main theorem
  15. Degrees of irreducible characters
  16. ...and 13 more sections

Key Result

lemma 1

Suppose $\mu\in\calp$, and let $(\mu^{(0)},\mu^{(1)})$ be the $2$-quotient of $\mu$. Then the $2$-quotient of $\mu'$ is $((\mu^{(1)})',(\mu^{(0)})')$.

Figures (1)

  • Figure 1: The matrices $E$ and $EA^{-1}$ for a Rouquier block of weight $7$

Theorems & Definitions (18)

  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6
  • lemma 7
  • lemma 8
  • lemma 9
  • lemma 10
  • ...and 8 more