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

TL;DR

This paper explores arithmetic properties of spin characters of double covers of symmetric and alternating groups, deriving Ramanujan-like congruences through generating function analysis.

Contribution

It introduces new congruences for spin characters using classical analytic tools and generating function manipulations, expanding understanding of their arithmetic structure.

Findings

Derived Ramanujan-like congruences for spin characters

Connected generating functions to classical analytic methods

Enhanced understanding of the arithmetic of spin characters

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.