A polytope combinatorics for semisimple groups

TL;DR

This paper explores the combinatorial structure of polytopes arising from the intersection homology of loop Grassmannian strata, providing tools to compute representation multiplicities for semisimple groups.

Contribution

It introduces a polytope combinatorics framework for semisimple groups, linking geometric cycles to representation theory computations.

Findings

Explicit descriptions of polytopes for low rank groups

Method to compute weight multiplicities using moment map images

Connection between polytopes and tensor product multiplicities

Abstract

Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they may be used to compute weight multiplicities and tensor product multiplicities for representations of a semisimple group. The polytopes are explicitly described for a few low rank groups.