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Erschienen in: Applicable Algebra in Engineering, Communication and Computing 4/2022

08.09.2020 | Original Paper

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

verfasst von: Rebecca E. Garcia, Pamela E. Harris, Marissa Loving, Lucy Martinez, David Melendez, Joseph Rennie, Gordon Rojas Kirby, Daniel Tinoco

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 4/2022

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Abstract

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.

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Metadaten
Titel
On Kostant’s weight q-multiplicity formula for
verfasst von
Rebecca E. Garcia
Pamela E. Harris
Marissa Loving
Lucy Martinez
David Melendez
Joseph Rennie
Gordon Rojas Kirby
Daniel Tinoco
Publikationsdatum
08.09.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing / Ausgabe 4/2022
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00454-8

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