ePrints@IIScePrints@IISc Home | About | Browse | Latest Additions | Advanced Search | Contact | Help

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).

[img] PDF
sem_lot_com_86_2022.pdf - Published Version
Restricted to Registered users only

Download (175kB) | Request a copy
Official URL: https://doi.org/10.48550/arXiv.2206.04509


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
Publication: Seminaire Lotharingien de Combinatoire
Publisher: Universitat Wien, Fakultat fur Mathematik
Additional Information: The copyright for this article belongs to the Universitat Wien, Fakultat fur Mathematik.
Keywords: 212-closed subsets; highest weight modules; Root systems; weak faces
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 11 Jul 2023 06:48
Last Modified: 11 Jul 2023 06:48
URI: https://eprints.iisc.ac.in/id/eprint/82425

Actions (login required)

View Item View Item