Kostant’s weight q-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. The q-analog of the partition function is a polynomial-valued function defined by \(\wp _q(\xi )=\sum _{i=0}^k c_i q^i\), where \(c_i\) is the number of ways the weight \(\xi\) can be written as a sum of exactly i positive roots of a Lie algebra \({\mathfrak {g}}\). The evaluation of the q-multiplicity formula at \(q = 1\) recovers the multiplicity of a weight in an irreducible highest weight representation of \({\mathfrak {g}}\). In this paper, we specialize to the Lie algebra \({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\) and we provide a closed formula for the q-analog of Kostant’s partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of \({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\)), and use these results to provide a closed formula for the q-multiplicity for any pair of dominant integral weights of \({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\). Throughout this work, we provide code to facilitate these computations.

Kostant 的权重q -multiplicity 公式是在称为 Weyl 群的有限群上的交替和，其项涉及Kostant 配分函数的q -类比。配分函数的q类比是由\(\wp _q(\xi )=\sum _{i=0}^k c_i q^i\)定义的多项式值函数，其中\(c_i\)是权重\(\xi\)可以写成李代数\({\mathfrak {g}}\)的正好i个正根的总和的方式数。在\(q = 1\)处对q -multiplicity 公式的评估恢复了权重在不可约的最高权重表示中的多重性\({\mathfrak {g}}\)。在本文中，我们专门研究李代数\({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\)，并为q - Kostant 分区函数的模拟，它扩展了 Shahi、Refaghat 和 Marefat 的最新结果。我们还描述了多重性公式的支持集（称为\({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\) 的 Weyl 交替集） )，并使用这些结果为\ ({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb { C}})\)。在整个工作中，我们提供代码来促进这些计算。