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On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$

Arzu Boysal

Abstract

For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $ρ$ be the half sum of positive roots and $P(2ρ)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $λ$ of the form $λ=2ρ-\sum_{i=1}^r a_iα_i$ with $a_i\in \mathbb{Z}_{\geq 0}$ for $1\leq i\leq r$. We show that if $λ$ is a vertex of $P(2ρ)$, then $V(λ)$ appears in $V(ρ) \otimes V(ρ)$ with multiplicity one, proving partially (for the vertices of $P(2ρ)$) a conjecture of Kostant describing components of $V(ρ)\otimes V(ρ)$. This result allows us to give an alternative proof for a weaker form of the conjecture (up to saturation factor) for any $\mathfrak{g}$. Further, using works of Knutson-Tau on the saturation property of $\mathfrak{sl_{r+1}}$, our results give an alternative proof of Kostant's conjecture in the particular case $\mathfrak{g}=\mathfrak{sl_{r+1}}$.

On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$

Abstract

For a complex simple Lie algebra or rank , let be the half sum of positive roots and be the convex hull of all dominant weights of the form with for . We show that if is a vertex of , then appears in with multiplicity one, proving partially (for the vertices of ) a conjecture of Kostant describing components of . This result allows us to give an alternative proof for a weaker form of the conjecture (up to saturation factor) for any . Further, using works of Knutson-Tau on the saturation property of , our results give an alternative proof of Kostant's conjecture in the particular case .
Paper Structure (9 sections, 7 theorems, 26 equations)

This paper contains 9 sections, 7 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminary constructions and vertices of dominant weight polytopes
  3. Preleminaries
  4. Characterization of vertices of dominant weight polytopes
  5. Proof of Theorem \ref{['th:main3']}
  6. Character of $V(\rho)\otimes V(\rho)$, its relation to $P(2\rho)$ and proof of Theorem \ref{['th:main1']}
  7. Characterization of weights in $P(2\rho)$
  8. The proof of Theorem \ref{['th:main1']}
  9. Saturation and the proof of Theorem \ref{['th:main2']}

Key Result

Theorem 1.2

Suppose $\mu$ is a regular dominant integral weight of $\mathfrak{g}$. Denote by $P(\mu)$ the polytope given by the intersection of the convex hull of the Weyl group orbit $W\cdot \mu$ and the dominant rational cone. Then, the vertices of $P(\mu)$ are in bijective correspondence with subsets $J\subs where $W_J$ is the subgroup of the Weyl group $W$ generated by simple reflections $s_{\alpha_i}\;(i

Theorems & Definitions (14)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof
  • Proposition 2.3
  • ...and 4 more