Related papers: Involution words II: braid relations and atomic st…

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and…

We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic…

An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function f:S->S that satisfies f(xy)=f(y)f(x) and f(f(x))=x for all x,y in S. The set I(S) of all such involutions on S generates a…

Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$.…

In this paper we consider a function $L(\vec{x})=L(a,b,c,d;e;f,g)$, which can be written as a linear combination of two Saalsch\"utzian ${}_4F_3(1)$ hypergeometric series or as a very-well-poised ${}_7F_6(1)$ hypergeometric series. We…

We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group $W$. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the…

The deletion order of a finitely generated Coxeter group W is a total order on the elements which, as is proved, is a refinement of the Bruhat order. This order is applied in [8] to construct Elnitsky tilings for any finite Coxeter group.…

Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a…

Weak Bruhat interval modules of the $0$-Hecke algebra in type $A$ provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were generalized from type $A$…

Let $w$ be a word in a free group. A few years ago, Magee and the first named author discovered that the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain Fourier coefficients…

Let $(W,S)$ be a Coxeter system and $\Gamma$ be a group of automorphisms of $W$ such that $\gamma(S)=S$ for all $\gamma \in \Gamma$. Then it is known that the group of fixed points $W^\Gamma$ is again a Coxeter group with a canonically…

Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite…

In one of his papers on the weak order of Coxeter groups, Dyer formulates several conjectures. Among these, one affirms that the extended weak order forms a lattice, while another offers an algebraic-geometric description of the join of two…

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…

It is known when we call a poset P, a $\mathcal{P}$-chain permutational poset, given a subset of permutations $\mathcal{P}$ of the symmetric group $S_{n}$. In this work, we use the same idea to study subsets of words of length $n$, that are…

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…

Iwahori-Hecke algebras are $q$-deformations of group algebras of Coxeter groups. In this article, we initiate a systematic study of quantum metric structures on Iwahori-Hecke algebras by establishing that, for finite rank right-angled…

According to an old result of Sch\"utzenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups.…

We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which…

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide…