Related papers: Conjugacy classes and straight elements in Coxeter…

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The…

A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this…

This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank $n$, which is the free product $W_n$ of $n$ copies of $\mathbb{Z}/2\mathbb{Z}$. We prove that for…

For a positive braid $\beta \in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(\beta)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(\beta)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater…

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This…

Let $G$ be a connected reductive group over an algebraically closed field with Weyl group $W$. The analogy between Lusztig varieties and Deligne-Lusztig varieties associated to minimal length elements in elliptic conjugacy classes of $W$…

Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set.…

We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras:…

For any permutation w, we characterize the reduced words of w that are their own commutation class. When w is the long element n(n-1)...321 and n \ge 4, there are exactly four such words.

Given a Coxeter system (W,S) equipped with an involutive automorphism T, the set of twisted identities is i(T) = {T(w)^{-1}w : w \in W}. We point out how i(T) shows up in several contexts and prove that if there is no s \in S such that…

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every…

Let W be a Weyl group. We introduce the notion of positive conjugacy class in W. This generalizes the notion of regular elliptic conjugacy class in the sense of Springer.

We prove a convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool we introduce new semigroup of partial permutations. We…

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have…

Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $\phi$ of $W$ over the finite field $F_2$ of two elements. The action of $\phi(W)$ on $F_2^n$ by left multiplication is…

Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact…

Artin groups are a natural generalization of braid groups and are well-understood in certain cases. Artin groups are closely related to Coxeter groups. There is a faithful representation of a Coxeter group $W$ as a linear reflection group…

We introduce and study "2-roots", which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram $Y_{a,b,c}$ having $n$…

One version of Whitehead's famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut…

The reflections in a Coxeter group are defined as conjugates of a single generator, and thus admit palindromic expressions as products of generators. Our main result gives closed formulas providing a palindromic reduced expression for each…