Lattice structure of Weyl groups via representation theory of preprojective algebras

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 $Π$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $τ$-rigid (respectively, $τ^-$-rigid) modules and layers of $Π$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $Π$. We show that layers of $Π$ 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 $τ^-$-rigid modules for type $A$ and $D$.