On Kostant's weight $q$-multiplicity formula for $\mathfrak{sl}_{4}(\mathbb{C})$

Abstract

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 sl4(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 sl4(C), which are associated to the Weyl alternation sets. This work answers a question posed in 2019

Weight $q$-multiplicities for representations of the exceptional Lie algebra $\mathfrak{g}_2$

“…In this section, we provide a closed formula for the q-analog of Kostant's partition function for the exceptional Lie algebra g 2 , which was presented in equation (3). We restate the result below for ease of reference.…”

“…In general, there has been some success in providing closed formulas for weight q-multiplicities for Lie algebras of low rank. This includes the work of Harris and Lauber [10] on weight q-multiplicities for the representations of sp 4 (C), which generalized the the work of Refaghat and Shahryari [14], and the work of Garcia, Harris, Loving, Martinez, Melendez, Rennie, Rojas Kirby, and Tinoco [3] on weight q-multiplicities for sl 4 (C). Other work provides visualizations of the subsets of elements of the Weyl group which contribute non-trivially to the associated weight multiplicity, for examples see [11,12].…”