2024/09/19 by G. Krishna Teja, Teja, G. Krishna
Environmental Science · Mathematics · #17B22 #17B67 #17B70 #52B20 #52B99 #FOS: Mathematics #Point processes and geometric inequalities #Primary: 17B10 #Representation Theory (math.RT) #Secondary: 17B20 #Statistical and numerical algorithms #Wind and Air Flow Studies
paper · pdf · doi:10.48550/arxiv.2409.12802
openalex publication_date 2024/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix any complex Kac-Moody Lie algebra \mathfrakg, and Cartan subalgebra \mathfrakh⊂ \mathfrakg. We study arbitrary highest weight \mathfrakg-modules V (with any highest weight λ∈ \mathfrakh^*, and let L(λ) be the corresponding simple highest weight \mathfrakg-module), and write their weight-sets wt V. This is based on and generalizes the Minkowski decompositions for all wt L(λ) and hulls convℝ(wt V), of Khare [J. Algebra. 2016 & Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 & J. Algebra. 2022]. Those works need a freeness property of the Dynkin graph nodes of integrability Jλ of L(λ): wt L(λ) - any sum of simple roots over Jλc are all weights of L(λ). We generalize it for all V, by introducing nodes JV that record all the lost 1-dim. weights in V. We show three applications (seemingly novel) for all (\mathfrakg, λ, V) of our JVc-freeness: 1) Minkowski decompositions of all wt V, subsuming those above for simples. 1') Characterization of these formulas. 1'') For these, we solve the inverse problem of determining all V with fixing wt V = weight-set of a Verma, parabolic Verma and L(λ) ∀ λ. 2) At module level (by raising operators' actions), construction of weight vectors along JVc-directions. 3) Lower bounds on the multiplicities of such weights, in all V.