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

TL;DR

This paper proves that certain vertices of a convex polytope associated with a simple Lie algebra correspond to unique components in the tensor product of the half-sum of positive roots, partially confirming Kostant's conjecture.

Contribution

It establishes that vertices of a specific convex hull correspond to unique tensor product components, providing partial proof of Kostant's conjecture and alternative proofs for special cases.

Findings

Vertices of P(2ρ) correspond to unique components in V(ρ)⊗V(ρ).

Provides an alternative proof of a weaker form of Kostant's conjecture.

Offers a new proof of Kostant's conjecture for sl_{r+1} using saturation results.

Abstract

For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $\rho$ be the half sum of positive roots and $P(2\rho)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $\lambda$ of the form $\lambda=2\rho-\sum_{i=1}^r a_i\alpha_i$ with $a_i\in \mathbb{Z}_{\geq 0}$ for $1\leq i\leq r$. We show that if $\lambda$ is a vertex of $P(2\rho)$, then $V(\lambda)$ appears in $V(\rho) \otimes V(\rho)$ with multiplicity one, proving partially (for the vertices of $P(2\rho)$) a conjecture of Kostant describing components of $V(\rho)\otimes V(\rho)$. 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 $\mathfrak{g}=\mathfrak{sl_{r+1}}$.