Higher Bruhat orders in type B

Seth Shelley-Abrahamson¹ ; Suhas Vijaykumar¹

¹Massachusetts Institute of Technology

The electronic journal of combinatorics, Tome 23 (2016) no. 3

Résumé

Motivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer $n \geq 1$ 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 $B(n, k)$. 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 $B_n$.

Détail

DOI : 10.37236/5620

Classification : 06A07, 06A11, 05E10, 20F55
Mots-clés : Coxeter theory, poset, Bruhat order