A facial order for torsion classes

Eric J. Hanson

Published 2023-05-10Version 1

Motivated by the "facial weak order" on posets of regions of hyperplane arrangements, we define a new partial order (the "binuclear interval order") on certain intervals in the lattice of torsion classes of an abelian length category. The binuclear interval order can more generally be formulated for any complete lattice satisfying a technical assumption, and it restricts to the facial weak order in the relevant setting. Over finite-dimensional algebras, we then consider the restriction of the binuclear interval order to those intervals coming from stability conditions, yielding what we call the "facial semistable order". We give two additional interpretations of (subposets of) the facial semistable order: one by describing its cover relations, and one in terms of Bongartz completions of 2-term presilting objects. For algebras which are $\tau$-tilting finite, we use these descriptions to prove that the facial semistable order is a semidistributive lattice. We then show that, over any abelian length category, the binuclear interval order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes.