Exceptional sequences in semidistributive lattices and the poset topology of wide subcategories

TL;DR

This paper explores the structure of semidistributive lattices and wide subcategories in representation theory, introducing new concepts like $ au$-exceptional sequences and $ ext{ extkappa}^d$-exceptional sequences, linking lattice theory with algebraic and combinatorial properties.

Contribution

It generalizes known results for finite semidistributive lattices, introduces $ ext{ extkappa}^d$-exceptional sequences, and connects lattice-theoretic concepts with algebraic structures like $ au$-tilting theory.

Findings

$ ext{ extkappa}$-map characterizes canonical join representations

Canonical join complex is a flag simplicial complex

$ ext{ extkappa}^d$-exceptional sequences generalize $ au$-exceptional sequences

Abstract

Let $\Lambda$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $\tau$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated $\Lambda$-modules. When $\Lambda$ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with a bijection between their completely join-irreducible and completely meet-irreducible elements, known as rowmotion or simply the "$\kappa$-map". Generalizing known results for finite semidistributive lattices, we show that the $\kappa$-map determines exactly when a set of completely join-irreducible elements forms a "canonical join representation". A consequence is that the corresponding "canonical join complex" is a flag simplicial complex, as has been shown for finite semidistributive lattices and lattices of torsion classes of finite-dimensional algebras. Finally, in the case of lattices of torsion classes of finite-dimensional algebras, we demonstrate how Jasso's $\tau$-tilting reduction can be encoded using the $\kappa$-map. We use this to define $\kappa^d$-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to $\tau$-exceptional sequences in the algebra setting.