Speaker
Description
In this minicourse, I will be focusing on the representation theory of lattices (that is to say, of their incidence algebras). This turns out to be an interesting setting in which to explore Auslander regularity and related notions, in part because they make contact with certain already studied notions within lattice theory, which I will also discuss. This minicourse assumes no prior knowledge of lattice theory, and the representation theory background required is fairly minimal.
I will explain the definition of Auslander regularity, and then present the result of Iyama and Marczinzik showing that a lattice is Auslander regular if and only if it is distributive. I will also explain how Auslander regularity (of incidence algebras of lattices) can be read off from the Coxeter matrix. This leads naturally to the notion of Coxeter permutation, and the rowmotion operation on distributive lattices. Finally, I will discuss the extension of rowmotion to semidistributive lattices, the way this manifests in terms of the Coxeter matrix, and how this can also be understood in representation-theoretic terms.