论文标题
重量$ q $ -mmultiplicities用于特殊谎言代数$ \ mathfrak {g} _2 $的表示形式
Weight $q$-multiplicities for representations of the exceptional Lie algebra $\mathfrak{g}_2$
论文作者
论文摘要
给定一个简单的谎言代数$ \ mathfrak {g} $，Kostant的重量$ Q $ -MMULTIPLICITICITICY公式是与Weyl组交替的总和，其术语涉及Kostant分区功能的$ Q $ -Analog。 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 $\mathfrak{g}$.这样，对Kostant的重量$ Q $ -Multiplicity公式的评估$ Q = 1 $恢复了重量的多重性，以$ \ Mathfrak {g} $的最高权重表示。在本文中，我们给出了用于计算权重$ q $ - 多重表示的封闭公式，以最高的重量表示代数代数$ \ 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 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 $\mathfrak{g}_2$.
