Characters and spin characters of alternating and symmetric groups determined by values on l'-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' = 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.