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

## TL;DR

This paper extends Regev's virtual character results to identify nonzero Kronecker coefficients for self-conjugate partitions, providing new insights into character theory and partition combinatorics.

## Contribution

It generalizes Regev's virtual character approach to self-conjugate partitions and demonstrates its application in identifying nonzero Kronecker coefficients.

## Key findings

- Established a method to find nonzero Kronecker coefficients for self-conjugate partitions.
- Applied the virtual character framework to specific examples, confirming its effectiveness.
- Provided a new perspective on character criteria for Kronecker coefficients.

## 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 $\chi^\lambda(\mu)\neq0$ we find another partition $\tau$ related to $\lambda$ such that $\chi^\tau(\mu)\neq0$ 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.