# Weak Faces and a Formula for Weights of Highest Weight Modules Via Parabolic Partial Sum Property for Roots

Teja, GK (2022) Weak Faces and a Formula for Weights of Highest Weight Modules Via Parabolic Partial Sum Property for Roots. In: Seminaire Lotharingien de Combinatoire (86).

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Official URL: https://doi.org/10.48550/arXiv.2206.04509

## Abstract

Let g 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 g-modules: (1) We provide a minimal description for the weights of all non-integrable simple highest weight g-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 g-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 g-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.

Item Type: Journal Article Seminaire Lotharingien de Combinatoire Universitat Wien, Fakultat fur Mathematik The copyright for this article belongs to the Universitat Wien, Fakultat fur Mathematik. 212-closed subsets; highest weight modules; Root systems; weak faces Division of Physical & Mathematical Sciences > Mathematics 11 Jul 2023 06:48 11 Jul 2023 06:48 https://eprints.iisc.ac.in/id/eprint/82425