Characters and projective characters of alternating and symmetric groups determined by values on $l'$-classes

TL;DR

This paper classifies pairs of characters of alternating and symmetric groups that agree on conjugacy classes of elements with orders not divisible by a fixed integer, revealing their structure and implications for modular representation theory.

Contribution

It identifies all pairs of irreducible characters matching on specific conjugacy classes for various groups, extending understanding of character correspondences and modular representations.

Findings

Pairs of characters are conjugate or associate, labeled by partitions with parameters divisible by l.

When l is prime, the l-modular decomposition matrix rows are mostly distinct.

Additional pairs of characters are found when l=3, indicating special cases.

Abstract

This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer $l$, for $l \neq 3$. We do the same for the double covers of the symmetric and alternating groups. The only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by $l$. When $l$ is prime, this implies that the rows of the $l$-modular decomposition matrix are distinct except for the rows labelled by these pairs. When $l=3$ we exhibit many additional examples of such pairs of characters.