Spin characters of symmetric and alternating groups which are proportional in characteristic 3

Abstract

Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.