A second reduction-type formula for the refined Littlewood--Richardson coefficients in type A

Abstract

For a permutation $w$ in the symmetric group $S_n$ and partitions $\lambda ,\mu ,\nu$ with at most $n$ parts, the refined Littlewood--Richardson (LR) coefficients $c_{\lambda ,\mu }^{\nu }(w)$ in type $A_n$ count the multiplicity of the irreducible polynomial representation $V(\nu )$ of the general linear algebra ${\mathfrak{gl}}_n(\mathbb{C})$ appearing in the decomposition of the Kostant--Kumar submodule $K(\lambda ,w,\mu )$ of the tensor product $V(\lambda )\otimes V(\mu )$ of two irreducible polynomial ${\mathfrak{gl}}_n(\mathbb{C})$-modules. In this paper, we establish a second reduction-type formula for $c_{\lambda ,\mu }^{\nu }(w)$, extending the second reduction formula for the classical Littlewood--Richardson coefficients $c_{\lambda ,\mu }^{\nu }$. The proof relies on the hive model.

arXiv comment12 pages