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december 2020 Weight $q$-multiplicities for representations of the exceptional Lie algebra $\mathfrak{g}_2$
Jerrell Cockerham, Melissa Gutiérrez González, Pamela E. Harris, Marissa Loving, Amaury V. Miniño, Joseph Rennie, Gordon Rojas Kirby
Bull. Belg. Math. Soc. Simon Stevin 27(5): 641-662 (december 2020). DOI: 10.36045/j.bbms.200317

Abstract

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\linebreak Lie algebra $\mathfrak{g}_2$.

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Jerrell Cockerham. Melissa Gutiérrez González. Pamela E. Harris. Marissa Loving. Amaury V. Miniño. Joseph Rennie. Gordon Rojas Kirby. "Weight $q$-multiplicities for representations of the exceptional Lie algebra $\mathfrak{g}_2$." Bull. Belg. Math. Soc. Simon Stevin 27 (5) 641 - 662, december 2020. https://doi.org/10.36045/j.bbms.200317

Information

Published: december 2020
First available in Project Euclid: 24 December 2020

MathSciNet: MR4194215
Digital Object Identifier: 10.36045/j.bbms.200317

Subjects:
Primary: 17B10

Rights: Copyright © 2020 The Belgian Mathematical Society

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Vol.27 • No. 5 • december 2020
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