The Kronecker product of Schur functions indexed by two-row shapes or hook shapes

TL;DR

This paper derives simplified, closed-form formulas for Kronecker coefficients involving Schur functions indexed by two-row or hook shapes, using Sergeev's formula and comultiplication expansion, improving upon previous methods.

Contribution

The paper introduces a more natural and symmetric approach to compute Kronecker coefficients for specific shapes, providing simpler formulas than prior work.

Findings

Closed formulas for Kronecker coefficients with two-row or hook shapes

Use of Sergeev's formula and comultiplication expansion

Formulas are simpler and more symmetric than previous results

Abstract

The Kronecker product of two Schur functions $s_{\mu}$ and $s_{\nu}$, denoted by $s_{\mu}*s_{\nu}$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions $\mu$ and $\nu$. The coefficient of $s_{\lambda}$ in this product is denoted by $\gamma^{\lambda}_{{\mu}{\nu}}$, and corresponds to the multiplicity of the irreducible character $\chi^{\lambda}$ in $\chi^{\mu}\chi^{\nu}.$ We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for $s_{\lambda}[XY]$ to find closed formulas for the Kronecker coefficients $\gamma^{\lambda}_{{\mu}{\nu}}$ when $\lambda$ is an arbitrary shape and $\mu$ and $\nu$ are hook shapes or two-row shapes. Remmel \cite{Re1, Re2} and Remmel and Whitehead \cite{Re-Wh} derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach. We believe that the approach of this paper is more natural. The formulas obtained are simpler and reflect the symmetry of the Kronecker product.