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Congruences for spin characters of the double covers of the symmetric and alternating groups

Rishi Nath, James A. Sellers

Abstract

Let $p$ be an odd prime. The bar partitions with sign and $p$-bar-core partitions with sign respectively label the spin characters and $p$-defect zero spin characters of the double cover of the symmetric group, and by restriction, those of the alternating group. The generating functions for these objects have been determined by J. Olsson. We study these functions from an arithmetic perspective, using classical analytic tools and elementary generating function manipulation to obtain many Ramanujan-like congruences.

Congruences for spin characters of the double covers of the symmetric and alternating groups

Abstract

Let be an odd prime. The bar partitions with sign and -bar-core partitions with sign respectively label the spin characters and -defect zero spin characters of the double cover of the symmetric group, and by restriction, those of the alternating group. The generating functions for these objects have been determined by J. Olsson. We study these functions from an arithmetic perspective, using classical analytic tools and elementary generating function manipulation to obtain many Ramanujan-like congruences.

Paper Structure

This paper contains 17 sections, 24 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and bar partitions
  4. Bar-core partitions and the bar-abacus
  5. Character theory
  6. Generating functions for $\hat{S}(n)$
  7. Generating functions for $\hat{A}(n)$
  8. Generating Function Manipulation Tools
  9. Congruences for spin characters
  10. Spin characters of $\hat{S}_n$
  11. Spin characters of $\hat{A}(n)$
  12. Arithmetic Results on $p$-bar core partitions
  13. Ramanujan--like congruences for $\bar{p}$-core partitions
  14. $p$-defect-zero spin characters of $\hat{S}_n$
  15. $p$-defect zero spin characters of $\hat{A}(n)$
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

For all $n\geq 0$,

Figures (3)

  • Figure 1: $S(\lambda)$ with bar lengths for $\lambda=(5,3,2)$
  • Figure :
  • Figure :

Theorems & Definitions (44)

  • Theorem \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 34 more