Related papers: Coxeter Pop-Tsack Torsing

4-dimensional $A_{4}$ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(A_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an…

We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the…

We study the triplet vertex operator algebra $\mathcal{W}(p)$ of central charge $1-\frac{6(p-1)^2}{p}$, $p \geq 2$. We show that $\trip$ is $C_2$-cofinite but irrational since it admits indecomposable and logarithmic modules. Furthermore,…

Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As…

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we…

We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a…

A referee found an error in the proof of the Theorem 2 that we could not fix. More precisely, the proof of Lemma 2.1 is incorrect. Hence the fact that integer cohomology of complement of toric Weyl arrangements is torsion free is still a…

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and…

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…

For a Coxeter group (W,S), a permutation of the set S is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter…

Let $(W,S)$ be a Coxeter system, let $S=I \dot{\cup} J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\widetilde{W}$ be…

It has been known that the centralizer $Z_W(W_I)$ of a parabolic subgroup $W_I$ of a Coxeter group $W$ is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathcal{Y}$. In this paper,…

The action of a Coxeter group $W$ on the set of left cosets of a standard parabolic subgroup deforms to define a module $\mathcal{M}^J$ of the group's Iwahori-Hecke algebra $\mathcal{H}$ with a particularly simple form. Rains and Vazirani…

We state combinatorial formulas for hyperoctahedral group ($\mathfrak B_n$) character evaluations of the form $\chi( {{\widetilde C}_w}^{\negthickspace\negthickspace BC}\negthickspace(1))$, where ${{\widetilde…

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By…

As a visualization of Cartier and Foata's "partially commutative monoid" theory, G.X. Viennot introduced "heaps of pieces" in 1986. These are essentially labeled posets satisfying a few additional properties. They naturally arise as models…

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular…

4-dimensional $F_{4} $ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an…

We consider the problem to determine the optimal rotations $R \in {\rm SO}(n)$ which minimize $$W: {\rm SO}(n) \to \mathbb{R}^+_0,\quad W(R\,;D) := ||{\rm sym}(RD - 1)||^2$$ for a given diagonal matrix $D := {\rm diag}(d_1, ..., d_n) \in…

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 $n$…