Lattice associated to a Shi variety

TL;DR

This paper explores the structure of the set of irreducible components of the Shi variety associated with an affine Weyl group, revealing it forms a semidistributive lattice, thus linking algebraic geometry and lattice theory.

Contribution

It demonstrates that the set of irreducible components of the Shi variety has a semidistributive lattice structure, a novel connection between geometric and algebraic structures.

Findings

The set of irreducible components forms a semidistributive lattice.

Establishes a new link between affine Weyl groups and lattice theory.

Provides a geometric interpretation of algebraic structures.

Abstract

Let $W$ be a irreducible Weyl group and $W_a$ its affine Weyl group. In a previous article the author defined 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 $\widehat{X}_{W_a}$, denoted $H^0(\widehat{X}_{W_a})$, is of some interest and we show in this article that $H^0(\widehat{X}_{W_a})$ has a structure of semidistributive lattice.