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On components of the tensor square of a Weyl module

Shiliang Gao, Dinglong Wang

Abstract

For a simple Lie algebra $\mathfrak{g}$ of type $A_n,B_n,C_n$ or $D_n$, we give a characterization of the set of dominant integral weights $λ$ such that for any rational point $μ$ in the fundamental Weyl chamber, $2λ-μ$ is a non-negative rational combination of the simple roots if and only if $V_{mμ}\subseteq V_{mλ}\otimes V_{mλ}$ for some positive integer $m$.

On components of the tensor square of a Weyl module

Abstract

For a simple Lie algebra of type or , we give a characterization of the set of dominant integral weights such that for any rational point in the fundamental Weyl chamber, is a non-negative rational combination of the simple roots if and only if for some positive integer .
Paper Structure (8 sections, 10 theorems, 46 equations, 1 figure)

This paper contains 8 sections, 10 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. notation and background
  3. Root system, weights and Weyl groups
  4. Dominant weight polytopes and their vertices
  5. Embedding of Dynkin diagrams and Schubert structure constants
  6. Inclusion of Dynkin diagrams and Schubert structure constants
  7. Inequalities defining the saturated tensor cones
  8. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.3

$\lambda$ satisfy eqn:mainQ if and only if

Figures (1)

  • Figure 1: Identification of Dynkin diagrams of type $A_3$ and $D_3$

Theorems & Definitions (17)

  • Conjecture 1.1: Kostant
  • Theorem 1.3
  • Theorem 2.1: BJK21, Proposition 3.9
  • Corollary 2.2: BJK21, Corollary 3.11
  • Lemma 2.3: BZ88, Proposition 1.3
  • Proposition 2.4: ABCD, Theorem 2.1
  • Remark 2.5
  • Proposition 2.6: BS00
  • Theorem 2.7: Horn inequalities
  • Theorem 2.8
  • ...and 7 more