A Facial Order for Torsion Classes

Author:

Hanson Eric J1

Affiliation:

1. Département de Mathématiques , LACIM, Université du Québec à Montréal and Département de Mathématiques, Université de Sherbrooke

Abstract

Abstract We generalize the “facial weak order” of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting “facial semistable order”: one using cover relations, and one using Bongartz completions of 2-term presilting objects. For $\tau $-tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes.

Funder

Canada Research Chairs program

Natural Sciences and Engineering Research Council of Canada

Publisher

Oxford University Press (OUP)

Reference57 articles.

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5. Classes of Semidistributive Lattices

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1. Correction to: A Facial Order for Torsion Classes;International Mathematics Research Notices;2024-09-03

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