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

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 $c^{\nu}_{\lambda,\mu}(w)$ 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 $c^{\nu}_{\lambda,\mu}(w)$, extending the second reduction formula for the classical Littlewood--Richardson coefficients $c^{\nu}_{\lambda,\mu}$. The proof relies on the hive model.