Realizations of Weyl groups on ellipsoids

TL;DR

This paper explores two geometric realizations of Weyl groups acting on solutions to Diophantine equations defining ellipsoids associated with complex semisimple Lie algebras, aiding in understanding their algebraic structure.

Contribution

It introduces two novel realizations of Weyl groups via Diophantine solutions on ellipsoids, facilitating analysis of the Bruhat order and reduced expressions.

Findings

Primary realization provides an order on the Weyl group.

Secondary realization helps find all reduced expressions for Weyl group elements.

Both realizations connect algebraic and geometric perspectives of Weyl groups.

Abstract

For any finite-dimensional complex semisimple Lie algebra two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group. The primary realization suggests an order on the Weyl group, which is useful for an exploration of the Bruhat order. The secondary realization is useful for finding for any element of the Weyl group all of its reduced expressions, which implies another realization of the Bruhat order.