Mathematics

The fundamental theorem of finite semi-distributive lattices

Hugh Thomas   [email] (Université du Québec à Montréal)

Abstract: The fundamental theorem of finite distributive lattices of Birkhoff says that any finite distributive lattice can be realized as the set of order ideals of a poset, ordered by inclusion. Semidistributive lattices are a generalization of distributive lattices, introduced by Jónsson in the 60s; he showed that free lattices are semidistributive. Among the interesting examples of finite semidistributive lattices are weak order on finite Coxeter groups and the torsion classes of an algebra (supposing there are only finitely many). I will present a theorem characterizing finite semidistributive lattices, formally similar to the fundamental theorem of finite distributive lattices. In a sense, this is a combinatorialization of the structure of torsion classes, but our construction does not actually use any representation theory, and I will not assume any knowledge of representation theory in my talk. This talk is based on arXiv:1907.08050, joint with Nathan Reading and David Speyer. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.