论文标题
精制的Littlewood-Richardson系数的饱和特性
The saturation property for refined Littlewood-Richardson coefficients
论文作者
论文摘要
给定有限维的简单谎言代数$ \ mathfrak {g} $和其Weyl oft的元素$ w $的主要积分重量$λ，μ，μ，ν$，精制的张量产物$ c_ {λμ}^ν（w）所谓的Kostant-Kumar Subsodule $ K（λ，W，μ）$ v（λ）\ otimes v（μ）$。我们在一般类型中得出这些系数的性能，包括brauer- klimyk型公式和限制定理。在类型$ a $中，我们获得了$ c_ {λμ}^ν（w）$的蜂巢模型，并证明如果置换$ W $为$ 312 $ - 避免$ 312 $，则饱和度和强的semigroup属性保持，$ 231 $ - 避免$ 231 $ - 避免使用此类元素的通勤产品。这概括了经典的Knutson---TAO饱和定理。
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.
