Chopped and sliced cones and representations of Kac-Moody algebras

TL;DR

This paper introduces chopped and sliced cones in combinatorial geometry, providing structure theorems for counting integral points, and applies these results to Kac-Moody algebra weight multiplicities and Littlewood-Richardson coefficients.

Contribution

It defines a new geometric concept and proves structure theorems, extending their application to important algebraic combinatorics problems.

Findings

Established structure theorems for integral points in sliced cones

Applied results to Kac-Moody algebra weight multiplicities

Extended framework to Littlewood-Richardson coefficients

Abstract

We introduce the notion of a chopped and sliced cone in combinatorial geometry and prove two structure theorems for the number of integral points in the individual slices of such a cone. We observe that this notion applies to weight multiplicities of Kac-Moody algebras and Littlewood-Richardson coefficients of semisimple Lie algebras, where we obtain the corresponding results.