A Combinatorial Interpretation for the coefficients in the Kronecker Product $s_{(n-p,p)}\ast s_λ$ (Multiplicities in the Kronecker Product $s_{(n-p,p)}\ast s_λ$)

Abstract

In this paper we give a combinatorial interpretation for the coefficient of $s_ν$ in the Kronecker product $s_{(n-p,p)}\ast s_λ$, where $λ=(λ_1, ..., λ_{\ell(λ)})\vdash n$, if $\ell(λ)\geq 2p-1$ or $λ_1\geq 2p-1$; that is, if $λ$ is not a partition inside the $2(p-1)\times 2(p-1)$ square. For $λ$ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all $λ$ when $n>(2p-2)^2$. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker products. We have also obtained some formulas for special cases.