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.