Higher Bruhat Orders in Type B

TL;DR

This paper extends the concept of higher Bruhat orders to type B, establishing their structure and connections to Weyl groups, and provides a partial generalization up to k=2.

Contribution

It introduces a partial generalization of higher Bruhat orders to type B, including a main theorem analogue and links to Weyl group structures.

Findings

Established a type B analogue of higher Bruhat orders.

Proved a main theorem similar to Manin and Schechtman's in type B.

Connected the construction to Weyl group orders and reduced expressions.

Abstract

Motivated by the geometry of certain hyperplane arrangements, Manin and Schechtman defined for each positive integer n 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 S_n, 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). 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 groups of type B.