Related papers: Mixed Bruhat operators and Yang-Baxter equations f…

Using the theory of PBW bases, one can realize the crystal $B(\infty)$ for any semisimple Lie algebra over $\mathbf{C}$ using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced…

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence.…

Let (W, S) be a Coxeter system. We investigate combinatorially certain partial orders, called extended Bruhat orders, on a (W x W)-set W(N,C), which depends on W, a subset N of S, and a component C of N. We determine the length of the…

We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak…

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 prime spectra of two families of algebras, $S^w$ and $\check{S}^w$, $w\in W,$ indexed by the Weyl group $W$ of a semisimple finitely dimensional are studied. The algebras $S^w$ have been introduced by A.~Joseph; they are $q$-analogues…

Suppose that $W$ is a finite Coxeter group and $W_J$ a standard parabolic subgroup of $W$. The main result proved here is that for any for any $w \in W$ and reduced expression of $w$ there is an Elnitsky tiling of a $2m$-polygon, where $m =…

Let $W$ be a finite reflection group. For a given $w \in W$, the following assertion may or may not be satisfied: (*) The principal Bruhat order ideal of $w$ contains as many elements as there are regions in the inversion hyperplane…

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…

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…

Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $\alpha\in R$, $v,w\in W$, we denote by $\partial_\alpha$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew…

Quantum Bruhat graph is a weighted directed graph on a finite Weyl group first defined by Brenti-Fomin-Postnikov. It encodes quantum Monk's rule and can be utilized to study the $3$-point Gromov-Witten invariants of the flag variety. In…

We extend the weak Bruhat order of a finite Coxeter group to the set of its coclasses, modulo parabolic standard subgroups. We use this order to describe associative algebra structures on the vector spaces spanned by the faces of…

We classify all quotients $W/W_J$ up to isomorphism in Bruhat order, with $(W,S)$ a Coxeter system and $W_J$ a parabolic subgroup of $W$. In particular, the non-trivial isomorphisms fall into a small number of cases which are highly…

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the $(d,m,l)$-generalized Yang-Baxter equation, for $m/2\leq l \leq m$, which allows to systematically…

Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of…

In this paper, we extend Manin and Schechtman's higher Bruhat orders for the symmetric group to higher Bruhat orders for non-longest words $w$ in $S_n$. We prove that the higher Bruhat orders of non-longest words are ranked posets with…

We introduce nil-Hecke algebras for Weyl groupoids. We describe a basis and some properties of these algebras which lead to a notion of Bruhat order for Weyl groupoids.

The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the…

The higher Bruhat orders are partial orders that generalize the weak order on the symmetric group $S_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in $S_n$, where covering…