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

• Main Authors: Ballantine, Cristina M, Orellana, Rosa C
• Format: Journal Article
• Language: English
• Published: 26.07.2005
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.math/0507544

S\'em. Lothar. Combin. 54A (2005/07), Art. B54Af In this paper we give a combinatorial interpretation for the coefficient of $s_{\nu}$ in the Kronecker product $s_{(n-p,p)}\ast s_{\lambda}$, where $\lambda=(\lambda_1, ..., \lambda_{\ell(\lambda)})\vdash n$, if $\ell(\lambda)\geq 2p-1$ or $\lambda_1\geq 2p-1$; that is, if $\lambda$ is not a partition inside the $2(p-1)\times 2(p-1)$ square. For $\lambda$ 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 $\lambda$ 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.