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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
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 𝔰𝔬4(), 𝔰𝔬5(), 𝔰𝔭4(), and the exceptional Lie algebra 𝔤2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on 𝔰𝔩3(), 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