Higher bruhat orders in type B

Abstract

Motivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer n ≥ 1 a hierarchy of finite partially ordered sets B(In; k); indexed by positive integers k, called the higher Bruhat orders. The poset B(In; 1) is naturally identified with the weak left Bruhat order on the symmetric group Sn, each B(In; k) has a unique maximal and a unique minimal element, and the poset B(In; k + 1) can be constructed from the set of maximal chains in B(In; k). Ben Elias has demonstrated a striking connection between the posets B(In; k) for k = 2 and the diagrammatics of Bott-Samelson bimodules in type A, providing significant motivation for the development of an analogous theory of higher Bruhat orders in other Cartan-Killing types, particularly for k = 2. In this paper we present a partial generalization to type B, completed up to k = 2, prove a direct analogue of the main theorem of Manin and Schechtman, and relate our construction to the weak Bruhat order and reduced expression graph for Weyl group Bn.