Convex Geometries on Root Systems

Abstract

This article investigates several convex closure operators on a finite root system. It is shown that a natural closure operator on the positive root system of a finite Weyl group satisfies the anti-exchange condition for all root systems except type F ₄.

Key Words:

• Convex geometries
• Coxeter groups
• Matroids
• Root systems

Mathematics Subject Classification:

• Primary 20F55, 52A01
• Secondary 17B20

ACKNOWLEDGMENT

I wish to thank M. Dyer for suggesting these questions and for some helpful conversations.

Notes

Communicated by M. Dixon.