Biclosed sets in Combinatorics

Abstract

The weak order is the set of permutations of [n] partially ordered by inclusion of inversion sets. This partial order arises naturally in various contexts, including enumer- ative combinatorics, hyperplane arrangements, Schubert calculus, cluster algebras, and many more. A fundamental result on the weak order is that the collection of maximal chains in any interval is connected by certain "local moves"�. Other notable features of the weak order are its lattice structure, its topology, and its geometry. The collection of inversion sets of permutations is an example of a family of biclosed sets. This thesis focuses on extending various nice properties of the weak order to other posets of biclosed sets. Some of these collections of biclosed sets have appeared previously in the literature, while others seem to be new. We briefly summarize our main results below. (�3.1.3) We give a criterion on a closure operator which ensures that the poset of biclosed sets is a congruence-uniform lattice. (�4) The chambers of a real simplicial or supersolvable hyperplane arrangement are in natural bijection with biclosed subsets of hyperplanes. (�4) completing the proof in [81]) The graph of reduced galleries of a supersolv- able hyperplane arrangement has diameter equal to the number of codimension 2 intersection subspaces. (�5) Chamber posets are semidistributive lattices if and only if they are crosscut- simplicial if and only if the arrangement is bineighborly. (�6) Every interval of the second Higher Bruhat order is either contractible or homotopy equivalent to a sphere. (�7) Every "facial"� interval of the poset of reduced galleries of a supersolvable arrangement is homotopy equivalent to a sphere. (�8) The Grid-Tamari orders are congruence-uniform lattices.