Higher Bruhat orders in type B (Q311505)

Summary: Motivated by the geometry of hyperplane arrangements, \textit{Yu. I. Manin} and \textit{V. V. Shekhtman} [Adv. Stud. Pure Math. 17, 289--308 (1989; Zbl 0759.20002)] defined for each integer \(n \geqslant 1\) 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 \(B(n, k)\). \textit{B. Elias} [Proc. Lond. Math. Soc. (3) 112, No. 5, 924--978 (2016; Zbl 1388.20011)] 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 [loc. cit.], and relate our construction to the weak Bruhat order and reduced expression graph for Weyl group \(B_n\).