On Kostant's conjecture for components of V(ρ)⊗ V(ρ)

2023/09/13 by Arzu Boysal, Boysal, Arzu
Mathematics · #17B10 #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2309.06890

openalex publication_date 2023/09/13 · openalex created_date 2023/09/15 · openalex updated_date 2026/07/28

Abstract

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

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