Higher Bruhat Orders Research Articles

Homotopy Type of Intervals of the Second Higher Bruhat Orders

The higher Bruhat order is a poset generalizing the weak order on permutations. Another special case of this poset is an ordering on simple wiring diagrams. For this case, we prove that every interval is either contractible or homotopy equivalent to a sphere. This partially proves a conjecture due to Reiner. Our proof uses some tools developed by Felsner and Weil to study wiring diagrams.

Higher Bruhat Orders in Type B

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 $B_n$.

Zonotopes associated with higher Bruhat orders

The higher Bruhat orders B(n,k) are combinatorially defined partial orders (and hence graphs) that “look like” the graphs of (n−k)-dimensional zonotopes—and they are for small parameters. Here we explain that this is since they contain the graphs of zonotopes of this dimension, but that in general they are not covered by these zonotopal graphs, and they are not polytopal in general. As a special case, this applies to the graph G n of all arrangements of n pseudolines connected by flips, since this graph is the graph of the higher Bruhat order B(n,2).

Higher Bruhat orders and relations in the braid groups

We prove relations in the braid group that generalize the relations of the formaba=bab. These relations are obtained by means of the generalization to the higher Bruhat orders of the well-known relation between the decompositions of the element of maximal length in the Weyl groupA n−1 and some specific linear orders on the system of positive roots ofA n−1 .

Lagrange interpolation with respect to Chebyshev systems and higher bruhat orders

Lagrange interpolation with respect to Chebyshev systems and higher bruhat orders

Triangulations of cyclic polytopes and higher Bruhat orders

Recently Edelman and Reiner suggested two poset structures, (n, d) and (n, d) on the set of all triangulations of the cyclic d-polytope C(n, d) with n vertices. Both posets are generalizations of the well-studied Tamari lattice. While (n, d) is bounded by definition, the same is not obvious for (n, d). In the paper by Edelman and Reiner the bounds of (n, d) were also confirmed for (n, d) whenever d≤5, leaving the general case as a conjecture.

Geometric constructions of higher Bruhat orders and M-Morsifications

Higher Bruhat orders were introduced by Manin and Schechtman in the course of studying multi-dimensional generalizations of the Yang-Baxter equations. In this paper we present a problem from real singularity theory which generalizes Arnol'ds snake calculus (a coding of the connected components of the space of very nice M-Morsifications of a singularity An) and in which the role of updown permutations is played by their higher analogues - elements of special form in higher Bruhat orders.

Higher bruhat orders and cyclic hyperplane arrangements

We study the higher Bruhat orders $B(n,k)$ of Manin & Schechtman [MaS] and - characterize them in terms of inversion sets, - identify them with the posets $U(C^{n+1,r},n+1)$ of uniform extensions of the alternating oriented matroids $C^{n,r}$ for $r:=n-k$ (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane), - show that $B(n,k)$ is a lattice for $k =1$ and for $r\le 3$, but not in general, - show that $B(n,k)$ is ordered by inclusion of inversion sets for $k=1$ and for $r\le 4$. However, $B(8,3)$ is not ordered by inclusion. This implies that the partial order $B_\subseteq (n,k)$ defined by inclusion of inversion sets differs from $B(n,k)$ in general. We show that the proper part of $B_\subseteq (n,k)$ is homotopy equivalent to $S^{r-2}$. Consequently, - $B(n,k)\simeq S^{r-2}$ for $k=1$ and for $r\le 4$. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for $r=3$, and that the proper part is not always homotopy equivalent to $S^{r(M)-2}$.

Free n-category generated by a cube, oriented matroids, and higher Bruhat orders

Free n-category generated by a cube, oriented matroids, and higher Bruhat orders

Higher Bruhat orders, related to the symmetric group

Higher Bruhat orders, related to the symmetric group