2022/06/09 by G. Krishna Teja, Teja, G. Krishna · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2206.04509
openalex publication_date 2022/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrakg be a finite or an affine type Lie algebra over ℂ with root system Δ. We show a parabolic generalization of the partial sum property for Δ, which we term the parabolic partial sum property. It allows any root β involving (any) fixed subset S of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from S, with each partial sum also being a root. We show three applications of this property to weights of highest weight \mathfrakg-modules: (1)~We provide a minimal description for the weights of all non-integrable simple highest weight \mathfrakg-modules, refining the weight formulas shown by Khare [J. Algebra 2016] and Dhillon-Khare [Adv. Math. 2017]. (2)~We provide a Minkowski difference formula for the weights of an arbitrary highest weight \mathfrakg-module. (3)~We completely classify and show the equivalence of two combinatorial subsets - weak faces and 212-closed subsets - of the weights of all highest weight \mathfrakg-modules. These two subsets were introduced and studied by Chari-Greenstein [Adv. Math. 2009], with applications to Lie theory including character formulas. We also show (3') a similar equivalence for root systems.