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

Abstract

Let $Λ$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $τ$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated $Λ$-modules. When $Λ$ 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 "$κ$-map". Generalizing known results for finite semidistributive lattices, we show that the $κ$-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 $τ$-tilting reduction can be encoded using the $κ$-map. We use this to define $κ^d$-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to $τ$-exceptional sequences in the algebra setting.