A virtual character and nonzero Kronecker coefficients for self-conjugate partitions

Abstract

We generalize Regev's result on a virtual character of $S_n$. Suppose that $λ$ and $μ$ are integer partitions of $n$. For the associated irreducible character $χ^λ$ of $S_n$, when $χ^λ(μ)\neq0$ we find another partition $τ$ related to $λ$ such that $χ^τ(μ)\neq0$ by the virtual character. Applying this result, we obtain a class of nonzero Kronecker coefficients by Pak et al.'s character criterion. Moreover, we discuss the effectiveness of Pak et al.'s character criterion by a concrete example.

18 pages; It's a revision of the the previous manuscript entitled 'On character method for Kronecker coefficients of self-conjugate partitions'