On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$
On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$
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关于 $$mathfrak {sl}_{4}(mathbb {C})$$ 的 Kostant 权重 q 重数公式
DOI：
10.1007/s00200-020-00454-8
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Tinoco, Daniel
中科院分区：
文献类型：
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作者：
Garcia, Rebecca E.;Harris, Pamela E.;Loving, Marissa;Martinez, Lucy;Melendez, David;Rennie, Joseph;Rojas Kirby, Gordon;Tinoco, Daniel
Kostant的权重乘法公式的q-类比是一个有限群上的交替和，称为Weyl群，其项涉及Kostant的配分函数的q-类比。这个公式，当求值于时，给出了一个简单李代数的最高权值表示中权值的多重。本文考虑李代数，给出了Kostant权重的q-类似的封闭公式。这个公式依赖于以下两组结果。首先，我们通过计数有限制的彩色整数划分，给出了Kostant配分函数的q-模拟的封闭公式。这些公式，当评估时，恢复De Loera和Sturmfels的结果。其次，我们描述并列举了Weyl交替集，它由Weyl群的元素组成，这些元素对Kostant的权重多重性公式有重要贡献。在此基础上，引入了与Weyl交替集相关联的根格上的Weyl交替图。这件作品回答了哈里斯、拉文、拉米雷斯、雷尼、罗哈斯·柯比、托雷斯·达维拉和尤利西斯在2019年提出的一个问题。
Theq-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve theq-analog of Kostant’s partition function. This formula, when evaluated at, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebraand give closed formulas for theq-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for theq-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.