Even $(\bar{s}, \bar{t})$-core partitions and self-associate characters of $\tilde{S}_n$

TL;DR

This paper extends methods to count even partitions that are simultaneously ar{s}-core and ar{t}-core, and applies these results to determine self-associate characters of S_n with specific defect properties.

Contribution

It introduces an extended method to count certain core partitions and applies this to characterize self-associate characters of S_n with defect zero for two primes.

Findings

Count of even ar{s}-core and ar{t}-core partitions determined.

Number of self-associate characters of S_n with defect 0 for two primes calculated.

Extension of Olsson and Bessenrodt's method to new combinatorial and representation-theoretic contexts.

Abstract

A partition is a $\bar{s}$-core if it is the result of removing all of the $s$-bars from a partition. We extend a method of Olsson and Bessenrodt to determine the number of even partitions that are simultaneously $\bar{s}$-core and $\bar{t}$-core. When $p$ and $q$ are distinct primes, this also determines the number of self-associate characters of $\tilde{S}_n$ that are simultaneosly defect 0 for $p$ and $q$.