Séminaire Lotharingien de Combinatoire, 86B.66 (2022), 12 pp.
G. Krishna Teja
Weak Faces and a Formula for Weights of Highest Weight Modules
Via Parabolic Partial Sum Property for Roots
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.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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