Fundamental polytope for the Weyl group acting on a maximal torus of a compact Lie group

TL;DR

This paper constructs explicit fundamental polytopes for the action of the Weyl group on a maximal torus in compact Lie groups, using root data and minuscule weights, with implications for automorphism groups.

Contribution

It provides a new, self-contained proof of the fundamental polytope description and extends understanding of Weyl group actions on tori in compact Lie groups.

Findings

Explicit convex hull description of the fundamental polytope.

Intersection of half-spaces characterizing the polytope.

Insights into automorphism groups of extended Dynkin diagrams.

Abstract

We provide a fundamental domain for the action of the finite Weyl group on a maximal torus of a compact Lie group of the corresponding type. The general situation is reduced to the adjoint case and, from the perspective of root data, this problem can be rephrased by asking for a fundamental polytope for the action of the extended affine Weyl group on the (dual) toral subalgebra. We solve the problem in this second form. Using the theory of minuscule weights, we obtain a description of this fundamental polytope as a convex hull of explicit vertices, and as an intersection of closed half-spaces. The latter description was first obtained by Komrakov and Premet in 1984 but, as the present work is independent of that of Komrakov-Premet, we give a new self-contained proof of it. We also derive some consequences on the structure of automorphism groups of extended Dynkin diagrams.