论文标题
在Kostant的重量上$ Q $ -Multiplicity公式，用于$ \ Mathfrak {sl} _ {4}（\ Mathbb {C}）$
On Kostant's weight $q$-multiplicity formula for $\mathfrak{sl}_{4}(\mathbb{C})$
论文作者
论文摘要
Kostant的重量多重性公式的$ Q $ -Analog是一个有限群体（称为Weyl ofter）的交替总和，其术语涉及Kostant分区功能的$ Q $ -Analog。当以$ q = 1 $评估时，此公式给出了一个简单的lie代数的最高权重表示重量的多样性。在本文中，我们考虑了Lie代数$ \ Mathfrak {Sl} _4（\ Mathbb {C}）$，并为Kostant重量多样性的$ Q $ -Analog提供了封闭的公式。该公式取决于以下两组结果。首先，我们通过计算受限制的彩色整数分区来展示Kostant分区功能的$ Q $ -Analog的封闭公式。这些公式以$ q = 1 $进行评估时，请恢复de loera和sturmfels的结果。其次，我们描述和枚举Weyl交替集，这些集集由Weyl基团的元素组成，这些元素对Kostant的重量多样性公式无效。由此，我们在$ \ mathfrak {sl} _4（\ Mathbb {c}）$的根晶格上介绍Weyl交替图，该图与Weyl交替集相关联。这项工作回答了哈里斯，爱，拉米雷斯，雷尼，罗哈斯·柯比，托雷斯·达维拉和尤利斯的一个问题。
The $q$-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostant's partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $\mathfrak{sl}_4(\mathbb{C})$ and give closed formulas for the $q$-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostant's partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, 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 $\mathfrak{sl}_4(\mathbb{C})$, 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.
