Higher Bruhat Orders in Type B

  • Seth Shelley-Abrahamson
  • Suhas Vijaykumar
Keywords: Coxeter theory, Poset, Bruhat order

Abstract

Motivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer n1 a hierarchy of finite partially ordered sets B(n,k), indexed by positive integers k, called the higher Bruhat orders.  The poset B(n,1) is naturally identified with the weak left Bruhat order on the symmetric group Sn, each B(n,k) has a unique maximal and a unique minimal element, and the poset B(n,k+1) can be constructed from the set of maximal chains in B(n,k).  Ben Elias has demonstrated a striking connection between the posets B(n,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, complete 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.
Published
2016-07-22
Article Number
P3.13