Related papers: Bruhat intervals and parabolic cosets in arbitrary…

In 2011, Dyer published a series of conjectures on the weak order of Coxeter groups. One of these conjectures stated that the inversion set of the join of two elements in a Coxeter group is equal to some "closure" of the union of their…

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their…

Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $\chi$-linear bicrystals. In this paper, we…

In this article, we give a short algebraic proof that all closed intervals in a $\gamma$-Cambrian semilattice $\mathcal{C}_{\gamma}$ are trim for any Coxeter group $W$ and any Coxeter element $\gamma\in W$. This means that if such an…

We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we…

Let $\mathrm{Int}(n)$ denote the set of nonempty left weak Bruhat intervals in the symmetric group $\mathfrak{S}_n$. We investigate the equivalence relation $\overset{D}{\simeq}$ on $\mathrm{Int}(n)$, where $I \overset{D}{\simeq} J$ if and…

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 determine the largest difference between the number of atoms and number of coatoms of a Bruhat interval of $S_n$.

We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection of parabolic subgroups in Euclidean Braid groups $A[\tilde{A}_n]$ is again a parabolic subgroup. To that end, we use that the…

We investigate the intersection covolume of cross sections for probability preserving actions of a class of abelian groups without lattices, including $p$-adic groups and the group of finite adeles. We show that for cross sections with a…

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 discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the…

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note…

For a finite Coxeter group $W$ and $w$ an element of $W$ the `excess' of $w$ is defined to be $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, \; x^2 = y^2 = 1\}$ where $\ell$ is the length function on $W$. Here we investigate the…

We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals,…

Let $G$ be a connected reductive group over an algebraically closed field. Let $B$ be a Borel subgroup of $G$ and $W$ be the associated Weyl group. We show that for any $w \in W$ that is not contained in any standard parabolic subgroup of…

The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is isomorphic to a poset on commutation classes of reduced words…

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations…

Let $(W,S)$ be a finite Weyl group and let $w\in W$. It is widely appreciated that the descent set D(w)=\{s\in S | l(ws)<l(w)\} determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some…