The saturation property for refined Littlewood-Richardson coefficients
Given dominant integral weights $λ, μ, ν$ of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and an element $w$ of its Weyl group, the refined tensor product multiplicity $c_{λμ}^ν(w)$ is the multiplicity of the irreducible $\mathfrak{g}$-module $V(ν)$ in the so-called Kostant--Kumar submodule $K(λ, w, μ)$ of the tensor product $V(λ) \otimes V(μ)$. We derive properties of these coefficients in general type, including a Brauer--Klimyk type formula and restriction theorems. In type $A$, we obtain a hive model for the $c_{λμ}^ν(w)$ and prove that the saturation and strong semigroup properties hold if the permutation $w$ is $312$-avoiding, $231$-avoiding, or a commuting product of such elements. This generalizes the classical Knutson--Tao saturation theorem.