On Kostant’s Conjecture for Components of V(ρ)⊗V(ρ)
Algebras and Representation Theory, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Basım Tarihi: 2026
- Doi Numarası: 10.1007/s10468-026-10410-8
- Dergi Adı: Algebras and Representation Theory
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, MathSciNet, zbMATH, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Anahtar Kelimeler: Kostant's conjecture, Sum of fundamental weights, Tensor product decomposition, Saturated tensor cone
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Boğaziçi Üniversitesi Adresli: Evet
Özet
For a complex simple Lie algebra g of rank r, let ρ be the half sum of positive roots and P(2ρ)⊂Rr be the convex hull of all dominant weights λ of the form λ=2ρ-∑i=1raiαi with ai∈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 vertex set 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 g. Further, using works of Knutson-Tau on the saturation property of slr+1, our results give an alternative proof of Kostant’s conjecture in the particular case g=slr+1.