Related papers: Combinatorial representations of Coxeter groups ov…

In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$…

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…

Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter…

We determine a fundamental domain for the diagonal action of a finite Coxeter group $W$ on $V^{\oplus n}$, where $V$ is the reflection representation. This is used to give a stratification of $V^{\oplus n}$, which is respected by the group…

The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously…

We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to…

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions…

Let $W$ be a finite Coxeter group with Coxeter generating set $S=\{s_1,\ldots,s_n\}$, and $\rho$ be a complex finite dimensional representation of $W$. The characteristic polynomial of $\rho$ is defined as \begin{equation*}…

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…

We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}.…

For every quiver (valued) of finite representation type we define a finitely presented group called a picture group. This group is very closely related to the cluster theory of the quiver. For example, positive expressions for the Coxeter…

Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-1)…

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main focus is the set $\D$ of distinguished involutions in $W$, which was introduced by Lusztig in…

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a…

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…

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the…

4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(H4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary…

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 study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices,…

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The…