Lattice structure of Weyl groups via representation theory of preprojective algebras

TL;DR

This paper explores the lattice structure of Weyl groups through the lens of representation theory of preprojective algebras, establishing bijections between algebraic and combinatorial objects and providing new insights into their organization.

Contribution

It introduces a novel connection between lattice congruences of Weyl groups and the representation theory of preprojective algebras, including bijections and algebraic descriptions.

Findings

Bijections between join-irreducible congruences, elements, and modules.

Identification of layers of preprojective algebras as bricks.

Description of the forcing order via doubleton extension order.

Abstract

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\Pi$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\tau$-rigid (respectively, $\tau^-$-rigid) modules and layers of $\Pi$. The lattice-theoretically natural labeling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\Pi$. We show that layers of $\Pi$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\tau^-$-rigid modules for type $A$ and $D$.