Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostant's partition function. For $\xi$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(\xi)=\sum c_i q^i$ where $c_i$ is the number of ways $\xi$ can be written as a sum of $i$ positive roots of $\mathfrak{g}$. In this way, the evaluation of Kostant's weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight representation of $\mathfrak{g}$. In this paper, we give closed formulas for computing weight $q$-multiplicities in a highest weight representation of the exceptional Lie algebra $\mathfrak{g}_2$.

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

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 ℘q(ξ)=∑i=0kciqi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\wp _q(\xi )=\sum _{i=0}^k c_i q^i$$\end{document}, where ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} is the number of ways the weight ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi$$\end{document} can be written as a sum of exactly i positive roots of a Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {g}}$$\end{document}. The evaluation of the q-multiplicity formula at q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q = 1$$\end{document} recovers the multiplicity of a weight in an irreducible highest weight representation of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {g}}$$\end{document}. In this paper, we specialize to the Lie algebra sp6(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})$$\end{document} 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 sp6(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})$$\end{document}), and use these results to provide a closed formula for the q-multiplicity for any pair of dominant integral weights of sp6(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})$$\end{document}. Throughout this work, we provide code to facilitate these computations.

On Kostant’s weight q-multiplicity formula for $${{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})$$

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.

pdf/on-kostant-s-weight-q-multiplicity-formula-for-mathfrak-sl-4-m5t8emsy81.pdf

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

. Given a simple graph G , one can deﬁne a hyperplane arrangement called the G -Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with G • -parking functions. When G is a complete graph we recover the full Shi arrangement, and the Pak-Stanley labels give a bijection with ordinary parking functions. However, for proper subgraphs G ⊂ K n , while the Pak-Stanley labels still include every G • -parking function, some appear more than once. These repetitions of Pak-Stanley labels are a topic of interest in the study of G -Shi arrangements and G • -parking functions. Furthermore, G • -parking functions are connected to many other combinatorial objects (for example, superstable conﬁgurations in chip-ﬁring). In studying these repetitions, we can draw on existing results about these objects such as Dhar’s Burning Algorithm. Conversely, our results have implications for the study of these objects as well. The key insight of our work is the introduction of a combinatorial model called the Three Rows Game. Analyzing the histories of this game and the ways in which they can induce the same outcomes allows us to characterize the multiplicities of the Pak-Stanley labels. Using this model, we develop a classiﬁcation theorem for the multiplicities of the Pak-Stanley labels of the regions in the P n -Shi arrangement, where P n is the path graph on n vertices. Then, we generalize the Three Rows Game into the T -Three Rows Game. This allows us to study the multiplicities of the Pak-Stanley labels of the regions in T -Shi arrangements, where T is any tree. Finally, we discuss the possibilities and diﬃculties in applying our method to arbitrary graphs. In particular, we analyze multiplicities in the case when G is a cycle graph, and prove a uniqueness result for maximal G • -parking functions for all graphs using the Three Rows Game.

pdf/repetitions-of-pak-stanley-labels-in-g-shi-arrangements-27ty8px2.pdf

Repetitions of Pak-Stanley Labels in $G$-Shi Arrangements

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

Repetitions of Pak-Stanley Labels in $G$-Shi Arrangements

Best Paper Award in Memory of Jacques Calmet