2023/05/10 by Eric J. Hanson, Hanson, Eric J.
Computer Science · Mathematics · #06A07 #06D75 (secondary) #15E10 #16G20 #18E40 #52C99 (primary) #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2305.06031
openalex publication_date 2023/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 τ-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.