Exceptional sequences in Semidistributive Lattices and the Poset Topology of wide Subcategories

Let Formula: see text be a finite-dimensional algebra over a field. We describe how Buan and Marsh’s Formula: see text-exceptional sequences can be used to give a “brick labeling” of a certain poset of wide subcategories of finitely generated Formula: see text-modules. When Formula: see text 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 toward 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 the “Formula: see text-map.” Generalizing known results for finite semidistributive lattices, we show that the Formula: see text-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. Finally, we demonstrate how Jasso’s Formula: see text-tilting reduction of finite-dimensional algebras can be encoded using the Formula: see text-map. We use this to define Formula: see text-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to Formula: see text-exceptional sequences in the algebra setting.