Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

Abstract

For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of a module category over a preprojective algebra of Dynkin type. This paper aims at studying categorical properties of $\mathcal{F}(w)$ via its connection with the root system. We show that by taking dimension vectors, simple objects in $\mathcal{F}(w)$ bijectively correspond to Bruhat inversion roots of $w$. As an application, we obtain a combinatorial criterion for $\mathcal{F}(w)$ to satisfy the Jordan-Hölder property (JHP). To achieve this, we develop a method to find simple objects in a general torsion-free class by using a brick sequence associated to a maximal green sequence of it. For type A case, we give a diagrammatic construction of simple objects, and show that (JHP) can be characterized via a forest-like permutation, introduced by Bousquet-Mélou and Butler in the study of Schubert varieties.

arXiv comment30 pages, minor revision, comments welcome

References in corpus (12)

• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Kac-Moody groups and cluster algebras
• Noncrossing partitions and representations of quivers
• Wall and Chamber Structure for finite-dimensional Algebras
• On the weak order of Coxeter groups
• Preprojective algebras and c-sortable words
• Lattice structure of Weyl groups via representation theory of preprojective algebras
• The Jordan-Hölder property and Grothendieck monoids of exact categories
• A survey on maximal green sequences
• An algebraic approach to Harder-Narasimhan filtrations
• Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
• Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

Cited by in corpus (5)

• The Jordan-Hölder property and Grothendieck monoids of exact categories
• ICE-closed subcategories and wide $\tau$-tilting modules
• Monobrick, a uniform approach to torsion-free classes and wide subcategories
• PBW parametrizations and generalized preprojective algebras
• Morphisms and extensions between bricks over preprojective algebras of type A