Related papers: Coxeter groups, symmetries, and rooted representat…

Let $\Gamma$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $\Gamma$.Recall that, by a theorem of H{\'e}e and M\"uhlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat…

In this fourth part, (with the notations of the preceding parts) we make the following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical and $S$ is finite. Let $R:W\to GL(M)$ be a reducible reflection representation of…

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…

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…

In this fith part, (with the notations of the preceding parts) we make the following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical and $S$ is of cardinality $3$. Let $R:W\to GL(M)$ be a reducible reflection…

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the…

We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary Coxeter group $W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear paring without any…

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,…

Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial…

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…

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show…

In this short, elementary note we prove that if a faithful reflection representation of a Coxeter group preserves an orthant, then that Coxeter group is a product of symmetric groups acting on its natural permutation representation. We also…

Let $G$ be a discrete Coxeter group, $G^+$ its alternating subgroup and $\tilde{G}^+$ the spinor cover of $G^+$. A presentation of the groups $G^+$ and $\tilde{G}^+$ is proved for an arbitrary Coxeter system $(G,S)$; the generators are…

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…

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such…

Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a…

An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is…

In this paper, we study Coxeter systems with two-dimensional Davis-Vinberg complexes. We show that for a Coxeter group $W$, if $(W,S)$ and $(W,S')$ are Coxeter systems with two-dimensional Davis-Vinberg complexes, then there exists…

We use geometry of Davis complex of a Coxeter group to prove the following result: if G is an infinite indecomposable Coxeter group and $H\subset G$ is a finite index reflection subgroup then the rank of H is not less than the rank of G.…

Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $\varphi$ be a weight function which is invariant under $G$. Let $\varphi_G$ denote the weight function on $W^G$ obtained by restriction…