Orientation of alcoves in affine Weyl groups

TL;DR

This paper explores the orientation of alcoves in affine Weyl groups by linking it to cohomology and the structure of the Shi variety, providing algebraic equations to determine alcove orientation efficiently.

Contribution

It introduces a new perspective on alcove orientation using cohomology and the irreducible components of the Shi variety, connecting geometric and algebraic viewpoints.

Findings

Orientation of alcoves expressed via first cohomology group

Irreducible components of Shi variety relate to alcove orientation

Modular equations describe alcove orientation property

Abstract

Let $W$ be an irreducible Weyl group and $W_a$ its affine Weyl group. In a previous work the author introduced an affine variety $\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. The set of irreducible components of $\widehat{X}_{W_a}$ provided results at the intersection of group theory, combinatorics and geometry. In this article we express the notion of orientation of alcoves in terms of the first group of cohomogoly of $W$ and in terms of the irreducible components of the Shi variety. We also provide modular equations in terms of Shi coefficients that describe efficiently the property of having the same orientation.