A 型中精细化 Littlewood-Richardson 系数的第二个约化型公式
A second reduction-type formula for the refined Littlewood--Richardson coefficients in type A
AI总结：
本文研究对称群\(S_n\)中置换及分划相关的\(A_n\)型精细化 Littlewood-Richardson 系数\(c^{\nu}_{\lambda,\mu}(w)\)，通过蜂巢模型建立其第二个约化型公式，扩展了经典系数的约化公式。
AI中文摘要：
对于对称群\(S_n\)中的置换\(w\)以及至多\(n\)部分的分划\(\lambda,\mu,\nu\)，\(A_n\)型精细化 Littlewood-Richardson（LR）系数\(c^{\nu}_{\lambda,\mu}(w)\)计算一般线性代数\(\mathfrak{gl}_n(\mathbb{C})\)的不可约多项式表示\(V(\nu)\)在两个不可约多项式\(\mathfrak{gl}_n(\mathbb{C})\)-模\(V(\lambda) \otimes V(\mu)\)的 Kostant-Kumar 子模\(K(\lambda,w,\mu)\)分解中出现的重数。本文建立了\(c^{\nu}_{\lambda,\mu}(w)\)的第二个约化型公式，扩展了经典 Littlewood-Richardson 系数\(c^{\nu}_{\lambda,\mu}\)的第二个约化公式。证明依赖于蜂巢模型。
英文摘要：
For a permutation $w$ in the symmetric group $S_n$ and partitions $λ, μ, ν$ with at most $n$ parts, the refined Littlewood--Richardson (LR) coefficients $c^ν_{λ,μ}(w)$ in type $A_n$ count the multiplicity of the irreducible polynomial representation $V(ν)$ of the general linear algebra $\mathfrak{gl}_n(\mathbb{C})$ appearing in the decomposition of the Kostant--Kumar submodule $K(λ,w,μ)$ of the tensor product $V(λ) \otimes V(μ)$ of two irreducible polynomial $\mathfrak{gl}_n(\mathbb{C})$-modules. In this paper, we establish a second reduction-type formula for $c^ν_{λ,μ}(w)$, extending the second reduction formula for the classical Littlewood--Richardson coefficients $c^ν_{λ,μ}$. The proof relies on the hive model.
