Dynamical combinatorics and torsion classes

For finite semidistributive lattices the map kappa gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the kappa-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the kappa-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend kappa to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll in 2019. For hereditary algebras, we show that the extended kappa-map on torsion classes is essentially the same as Ringel's epsilon-map on wide subcategories. Also in the hereditary case, we relate the square of kappa to the Auslander-Reiten translation. Published by Elsevier B.V.