Toric partial orders

Abstract

Toric partial orders correspond to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets arise from finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. There are natural toric analogues of many standard features of ordinary partial orders, such as chains, antichains, intervals, transitivity, Hasse diagrams, linear extensions, total orders, morphisms, and order ideals. Most of these only become apparent when one looks at these objects geometrically. Toric posets arise naturally in a wide variety of contexts, from the study of cyclic reducibility and conjugacy in Coxeter groups, to the critical path method (CPM) for scheduling activities with periodicity in operations research. This talk will be a survey on toric posets and it will be filled with lots of colorful pictures. This is joint work with Mike Develin and Vic Reiner.