论文标题
弱面和最高重量模块的权重的公式
Weak faces and a formula for weights of highest weight modules, via parabolic partial sum property for roots
论文作者
论文摘要
令$ \ mathfrak {g} $为有限或仿射类型lie代数，$ \ mathbb {c} $带有root System $Δ$。我们显示了$δ$的部分总和属性的抛物线概括，我们将其称为抛物线分支属性。它允许任何涉及（任何）简单根的固定子集$ s $的根$β$写成有序的根总和，每个根都完全涉及一个来自$ s $的简单根，每个部分总和也是根。我们将此属性的三个应用显示为最高权重$ \ mathfrak {g} $ - 模块：（1）〜我们提供了所有不可融合的简单最高权重$ \ mathfrak {g} $ - 模块的权重的最小描述，从Algebra} 2016]和Dhillon-Khare [adv。数学。 2017]。 （2）〜我们为任意最高权重$ \ mathfrak {g} $模块的权重提供了Minkowski差异公式。 （3）〜我们完全对所有最高权重$ \ mathfrak {g} $ - 模块的权重的两个组合子集（弱面和212封闭的子集）完全分类并显示。 Chari-Greenstein引入并研究了这两个子集[Adv。数学。 2009年]，应用于谎言理论，包括角色公式。我们还显示（$ 3'$）对根系的类似等价。
Let $\mathfrak{g}$ be a finite or an affine type Lie algebra over $\mathbb{C}$ with root system $Δ$. We show a parabolic generalization of the partial sum property for $Δ$, which we term the parabolic partial sum property. It allows any root $β$ involving (any) fixed subset $S$ of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from $S$, with each partial sum also being a root. We show three applications of this property to weights of highest weight $\mathfrak{g}$-modules: (1)~We provide a minimal description for the weights of all non-integrable simple highest weight $\mathfrak{g}$-modules, refining the weight formulas shown by Khare [J. Algebra} 2016] and Dhillon-Khare [Adv. Math. 2017]. (2)~We provide a Minkowski difference formula for the weights of an arbitrary highest weight $\mathfrak{g}$-module. (3)~We completely classify and show the equivalence of two combinatorial subsets - weak faces and 212-closed subsets - of the weights of all highest weight $\mathfrak{g}$-modules. These two subsets were introduced and studied by Chari-Greenstein [Adv. Math. 2009], with applications to Lie theory including character formulas. We also show ($3'$) a similar equivalence for root systems.
