Combinatorics Seminar

Speaker: Aram Dermenjian

Title: The facial weak order

Abstract: We discuss the facial weak order, a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement which was first introduced on the braid arrangements by Krob, Latapy, Novelli, Phan and Schwer. We provide various characterizations of this poset including combinatorial, algebraic and geometric ones. We simultaneously show these characterizations in hyperplane arrangements and Coxeter groups. We then show that the facial weak order is in fact a lattice for hyperplane arrangements whose poset of regions is a lattice, generalizing a result by Björner, Edelman and Ziegler. We end by stating some properties on the facial weak order and ongoing research on this order using shards; a tool developed by Nathan Reading.