Weight $q$-multiplicities for representations of the exceptional Lie algebra $\mathfrak{g}_2$
| Autor: | Gordon Rojas Kirby, Jerrell Cockerham, Marissa Loving, Melissa Gutiérrez González, Pamela E. Harris, Joseph Rennie, Amaury V. Miniño |
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| Rok vydání: | 2020 |
| Předmět: |
Weyl group
Partition function (quantum field theory) General Mathematics 17B10 Multiplicity (mathematics) Function (mathematics) $q$-weight multiplicities Combinatorics symbols.namesake Simple (abstract algebra) $q$-analog of Kostant's partition function exceptional Lie algebra $\mathfrak{g}_2$ Lie algebra Reidemeister trace FOS: Mathematics symbols Mathematics - Combinatorics Combinatorics (math.CO) Representation Theory (math.RT) Mathematics::Representation Theory Mathematics - Representation Theory Mathematics |
| Zdroj: | Bull. Belg. Math. Soc. Simon Stevin 27, no. 5 (2020), 641-662 |
| ISSN: | 1370-1444 |
| DOI: | 10.36045/j.bbms.200317 |
| Popis: | 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 $\mathfrak{g}_2$. Comment: 17 pages, 1 figure, tables |
| Databáze: | OpenAIRE |
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