Related papers: Interval structures in the Bruhat and weak orders

We study the restriction of the strong Bruhat order on an arbitrary Coxeter group $W$ to cosets $x W_L^\theta$, where $x$ is an element of $W$ and $W_L^\theta$ the subgroup of fixed points of an automorphism $\theta$ of order at most two of…

I introduce a method to generate families of CSS codes with interesting code parameters. The object of study is Coxeter groups, both finite and infinite (reducible or not), and a geometrically motivated partial order of Coxeter group…

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.…

In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof…

We investigate a poset structure that extends the weak order on a finite Coxeter group $W$ to the set of all faces of the permutahedron of $W$. We call this order the facial weak order. We first provide two alternative characterizations of…

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the…

Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in a linearly ordered set. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions of…

I present an overview of the research I have conducted for the past ten years in algebraic, bijective, enumerative, and geometric combinatorics. The two main objects I have studied are the permutahedron and the associahedron as well as the…

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent…

We determine the sharp asymptotic scale of the probability that two uniformly random permutations are comparable in weak Bruhat order, showing that $\mathbb{P}(\sigma_1 \preceq_W \sigma_2)=\exp\Bigl(\bigl(-\tfrac12+o(1)\bigr)\,n\log…

The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein* over Z_2. We also…

We introduce Lehmer codes, with immersions in the Bruhat order, for several finite Coxeter groups, including all the classical Weyl groups. This allows to associate to each lower Bruhat interval of these groups a multicomplex whose…

The set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The lattice structure is generalized to the set of words on a fixed alphabet $\Sigma = \{ x, y, z, ... \}$, where each letter has a…

The purpose of the present paper is to give a realization of a cylindric diagram as a subset of root systems of type $A_{\kappa-1}^{(1)}$ and several characterization of its poset structure. Furthermore, the set of order ideals of a…

Let u and v be permutations on n letters, with u <= v in Bruhat order. A Bruhat interval polytope Q_{u,v} is the convex hull of all permutation vectors z = (z(1), z(2),...,z(n)) with u <= z <= v. Note that when u=e and v=w_0 are the…

In [E. Tsukerman and L. Williams, {\em Bruhat Interval Polytopes}, Advances in Mathematics, 285 (2015), 766-810] it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this…

In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly…

We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we…

This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the…

In [Journal of Pure and Applied Algebra {224} (2020), no 12, 106449], V. Mazorchuk and R. Mr{\dj}en (with some help by A. Hultman) prove that, given a Weyl group, the intersection of a Bruhat interval with a parabolic coset has a unique…