Demazure Formulas for Weight Polytopes
Mark A. Walton
Abstract The characters of simple Lie algebras are naturally decomposed into
lattice-polytope sums. The Brion formula for those polytope sums is remarkably
similar to the Weyl character formula. Here we start to investigate if other character
formulas have analogs for lattice-polytope sums, by focusing on the Demazure
character formulas. Using Demazure operators, we write expressions for the lattice
sums of the weight polytopes of rank-2 simple Lie algebras, and the rank-3 algebra
A 3 .
Keywords Lattice polytopes · Simple Lie algebras · Characters · Demazure
character formula
1 Introduction
The Brion formula [3, 4] for lattice-polytope sums is remarkably similar to the Weyl
formula for characters of simple Lie algebras [6, 11, 15]. As a consequence, the
expansion of Weyl characters in terms of lattice-polytope sums is natural and useful
[6, 11–13, 15, 16]. This polytope expansion of Lie characters is highly reminiscent
of the early work of Antoine and Speiser [2] and the recursive formulas found by
Kass [8].
Here we explore further the relation between Lie characters and lattice-polytope
sums. Other formulas exist for the characters—might corresponding formulas
describe lattice-polytope sums? 1
We focus on the Demazure character formulas [1, 5, 7], and use the Demazure
operators involved to write expressions for the lattice-polytope sums. So far, we
1 This question was already asked in [15].
M. A. Walton ()
Department of Physics & Astronomy, University of Lethbridge, Lethbridge, AB, Canada
e-mail: walton@uleth.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_27
287
Mark A. Walton
Abstract The characters of simple Lie algebras are naturally decomposed into
lattice-polytope sums. The Brion formula for those polytope sums is remarkably
similar to the Weyl character formula. Here we start to investigate if other character
formulas have analogs for lattice-polytope sums, by focusing on the Demazure
character formulas. Using Demazure operators, we write expressions for the lattice
sums of the weight polytopes of rank-2 simple Lie algebras, and the rank-3 algebra
A 3 .
Keywords Lattice polytopes · Simple Lie algebras · Characters · Demazure
character formula
1 Introduction
The Brion formula [3, 4] for lattice-polytope sums is remarkably similar to the Weyl
formula for characters of simple Lie algebras [6, 11, 15]. As a consequence, the
expansion of Weyl characters in terms of lattice-polytope sums is natural and useful
[6, 11–13, 15, 16]. This polytope expansion of Lie characters is highly reminiscent
of the early work of Antoine and Speiser [2] and the recursive formulas found by
Kass [8].
Here we explore further the relation between Lie characters and lattice-polytope
sums. Other formulas exist for the characters—might corresponding formulas
describe lattice-polytope sums? 1
We focus on the Demazure character formulas [1, 5, 7], and use the Demazure
operators involved to write expressions for the lattice-polytope sums. So far, we
1 This question was already asked in [15].
M. A. Walton ()
Department of Physics & Astronomy, University of Lethbridge, Lethbridge, AB, Canada
e-mail: walton@uleth.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_27
287