Polytope sums and Lie characters

Abstract

Abstract. A new application of polytope theory to Lie theory is presented. Exponential sums of convex lattice polytopes are applied to the characters of irreducible representations of simple Lie algebras. The Brion formula is used to write a polytope expansion of a Lie character, that makes more transparent certain degeneracies of weight-multiplicities beyond those explained by Weyl symmetry.

Demazure formula for $A_n$ Weyl polytope sums

“…Polytope expansion. The Brion formula ( 12) is remarkably similar to the Weyl character formula (5) [14,11,6]. It is therefore natural and fruitful to consider the polytope expansion of Lie characters [14,6,15]:…”

“…In the weight lattice of a simple Lie algebra, a Weyl polytope has the weights in an orbit of the Weyl group as its vertices. Applied to Weyl polytopes, the Brion theorem yields a formula that is remarkably similar to the Weyl character formula [6,14,11]. As a consequence, the polytope expansion of Weyl characters in terms of integer-point transforms is natural and useful [14,11,13,15,6,12].…”

“…Here we explore further the relation between Lie characters and integer-point transforms. Other formulas exist for the characters, and following [14], we are interested in finding similar expressions for the integer-point transforms of Weyl polytopes, herein called Weyl polytope sums. We focus on the Demazure character formula [5,1,7] and use the Demazure operators involved to write expressions for the Weyl polytope sums.…”

Demazure Formulas for Weight Polytopes

“…The Brion formula ( 7) is remarkably similar to the Weyl character formula, as written in (4) [6,15]. It is therefore natural, and fruitful, to consider the polytope expansion of Lie characters [6,15,16]:…”

“…Correspondingly, the longest element w L of the Weyl group can be written as a product of the reflections defined in (15):…”