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 $\Pi$. 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 labelling 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 $D$.

arXiv comment32 pages. Version 2: Expository changes only. Version 3: Final pre-publication version

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• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Kac-Moody groups and cluster algebras
• Lattice congruences of the weak order
• Lattice structure of torsion classes for path algebras
• Preprojective algebras and c-sortable words
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Cited by in corpus (9)

• Lattice theory of torsion classes: Beyond $\tau$-tilting theory
• Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
• Bigrassmannian permutations and Verma modules
• Exceptional sequences and spherical modules for the Auslander algebra of $k[x]/(x^t)$
• The fundamental theorem of finite semidistributive lattices
• Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank
• The category of a partitioned fan
• Morphisms and extensions between bricks over preprojective algebras of type A