Demazure weight polytopes, inequalities, and lattice points

Speaker: Sam Jeralds
Affiliation: University of Queensland

Abstract

For a semisimple complex Lie group G, the structure and characters of its irreducible highest-weight representations is well-known and settled. In particular, the set of weights of such a representation forms a convex polytope, the Weyl polytope, which is easy to describe via its vertices. In this talk, we extend this approach to Demazure modules, certain Borel submodules of the irreducible G-modules. Demazure modules and their characters occupy a distinguished position in the intersection of representation theory, geometry, and algebraic combinatorics. We make use of each of these perspectives to attach to them the associated "Demazure weight polytope,"  describe the structure of these polytopes in terms of both vertices and inequalities, and highlight the connection between points of these polytopes and weights of the associated module.