Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

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

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 so_4(C), so_5(C), sp_4(C), and the exceptional Lie algebra g_2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl_3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights λand μof the Lie algebras considered.

Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

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 so_4(C), so_5(C), sp_4(C), and the exceptional Lie algebra g_2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl_3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights λand μof the Lie algebras considered.

Paper Structure

This paper contains 29 sections, 7 theorems, 23 equations, 32 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Lie algebra of type B2
  4. Lie algebra of type C2
  5. Lie algebra of type D2
  6. Lie algebra of type G2
  7. Weyl alternation sets
  8. Lie algebra of type B2
  9. Lie algebra of type C2
  10. Lie algebra of type D2
  11. Lie algebra of type G2
  12. Weyl alternation diagrams
  13. Lie algebra of type B2
  14. Case mu equals n omega 1
  15. Case mu equals m omega 2
  16. ...and 14 more sections

Key Result

Lemma 3.1

Let $\lambda = c_1\varpi_1+c_2\varpi_2$ with $c_1, c_2 \in {\mathbb Z}$. Then $\lambda \in {\mathbb Z}{\alpha_1} \oplus {\mathbb Z}{\alpha_2}$ if and only if $c_2$ is divisible by $2$.

Figures (32)

  • Figure 1: Weyl alternation diagram for $\mathfrak{sl}_3(\mathbb{C})$ with $\mu=0$. Reproduced from HLM.
  • Figure 2: Weyl alternation diagrams for the Lie algebras of type $B_2$, $C_2$, $D_2$, and $G_2$, where $\mu = 0$. Reproduced from harristhesis.
  • Figure 3: The root system of $B_2$.
  • Figure 4: The root system of $C_2$.
  • Figure 5: The root system of $D_2$.
  • ...and 27 more figures

Theorems & Definitions (16)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Theorem \oldthetheorem: Weyl alternation sets of $\mathfrak{so}_{5}(\mathbb{C})$
  • proof
  • Lemma 3.2
  • proof
  • Theorem \oldthetheorem: Weyl alternation sets of $\mathfrak{sp}_{4}(\mathbb{C})$
  • proof
  • Lemma 3.3
  • ...and 6 more