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 $\lambda$ and $\mu$ are integer partitions of $n$. For the associated irreducible character ${\chi }^{\lambda }$ of $S_n$, when ${\chi }^{\lambda }(\mu )\ne 0$ we find another partition $\tau$ related to $\lambda$ such that ${\chi }^{\tau }(\mu )\ne 0$ 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'