# Convexity properties of the canonical S-graphs

## TL;DR

This paper explores the convexity properties of S-graphs related to Kashiwara crystals, providing insights into their extremal elements and applications to understanding the structure of Verma modules in Kac-Moody algebras.

## Contribution

It establishes convexity properties of S-graphs and characterizes their extremal elements, advancing the understanding of Kashiwara crystals and Verma modules.

## Key findings

- Extremal elements of convex sets are S-sets from S-graphs.
- Convexity properties help describe Kashiwara crystals precisely.
- Applications to the structure of Verma modules in Kac-Moody algebras.

## Abstract

Let n be a positive integer and c an n-tuple of natural numbers. A convex set in Euclidean n-space given by a family of linear relations in the elements of c and depending on their natural order is defined. The extremal elements of this convex set are shown to be the S-set obtained from the S-graph defined by c constructed in studying the Kashiwara crystal B associated to a Verma module for a Kac-Moody algebra. The result has applications to the precise description of B.