Characters and spin characters of alternating and symmetric groups determined by values on $l^\prime$-classes

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 $l^\prime = 3$. We do likewise for spin characters of the symmetric and alternating groups. We find that 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.

Keywords

characters, projective representations, alternating group, symmetric group, decomposition numbers

2010 Mathematics Subject Classification

20C15, 20C20, 20C30

Full Text (PDF format)

Received 28 February 2023

Received revised 9 March 2023

Accepted 20 March 2023

Published 1 June 2024