可视化秩二李代数的 Kostant 权重重数公式的支集

Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers

Consider the weight $λ$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the representation with highest weight $λ$. We prove that in Lie algebras of type $A$ and $B$, the number of contributing terms to the multiplicity of the zero-weight space in the representation with highest weight $λ$ is given by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number.

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

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'…

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

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 $\wp_q(ξ)=\sum_{i=0}^k c_i q^i$, where $c_i$ is the number of ways the weight $ξ$ 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…

The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula

Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra? We describe and enumerate the cardinalities of these sets (through linear ho…

Kostant 权重重数公式的支撑是弱 Bruhat 序中的序理想

The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

本文证明了经典单李代数的 Weyl 交错集是弱 Bruhat 序中的序理想，完整刻画了 $\mathfrak{sl}_{r+1}(\mathbb{C})$ 的相关交错集，并为证明 Harry 的 $q$-模拟猜想提供了计数基础。

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

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 $ξ$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(ξ)=\sum c_i q^i$ where $c_i$ is the number of ways $ξ$ 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 represen…