Visualizing the support of Kostant’s weight multiplicity formula for the rank-2 Lie algebras

Harris, Pamela E.; Loving, Marissa; Ramirez, Juan; Rennie, Joseph; Rojas Kirby, Gordon; Torres Davila, Eduardo; Ulysse, Fabrice O.

doi:10.2140/involve.2024.17.183

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Visualizing the support of Kostant's weight multiplicity formula for the rank-2 Lie algebras

Pamela E. Harris, Marissa Loving, Juan Ramirez, Joseph Rennie, Gordon Rojas Kirby, Eduardo Torres Davila and Fabrice O. Ulysse

Vol. 17 (2024), No. 2, 183–215

DOI: 10.2140/involve.2024.17.183

Abstract

The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra $𝔤$ can be computed via Kostant’s weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant’s partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group, which contributes nontrivially to the multiplicity of a weight in a highest-weight representation of the Lie algebras ${𝔰 𝔬}_{4} (ℂ)$ , ${𝔰 𝔬}_{5} (ℂ)$ , ${𝔰 𝔭}_{4} (ℂ)$ , and the exceptional Lie algebra $𝔤_{2}$ . By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on ${𝔰 𝔩}_{3} (ℂ)$ , to provide visualizations of these Weyl alternation sets for all pairs of integral weights $λ$ and $μ$ of the Lie algebras considered.

Keywords

Kostant's partition function, weight multiplicity, Weyl alternation sets

Mathematical Subject Classification 2010

Primary: 05E10

Milestones

Received: 2 March 2020

Revised: 6 July 2022

Accepted: 19 February 2023

Published: 20 May 2024

Communicated by Kenneth S. Berenhaut

Authors

Pamela E. Harris
Department of Mathematics and Statistics Williams College Williamstown, MA United States

Marissa Loving
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

Juan Ramirez
Department of Mathematics University of Houston Houston, TX United States

Joseph Rennie
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

Gordon Rojas Kirby
Department of Mathematics University of California Santa Barbara, CA United States

Eduardo Torres Davila
Department of Mathematics San Diego State University San Diego, CA United States

Fabrice O. Ulysse
Department of Mathematics Cornell University Ithaca, NY United States

Vol. 17, No. 2, 2024