相关论文: Geometry of certain finite Coxeter group actions

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 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 $W$ be a finite Coxeter group. We classify the reflection subgroups of $W$ up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup $R$ of $W$ the conjugacy class of its Coxeter…

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for…

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…

Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine…

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail…

To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ and its connected…

We follow the dual approach to Coxeter systems and show for Weyl groups a criterium which decides whether a set of reflections is generating the group depending on the root and the coroot lattice. Further we study special generating sets…

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…

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…

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of…

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 present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody algebras. These domains are given as subsets of certain generalized upper half planes,…

H.S.M. Coxeter showed that a group $\Gamma$ is a finite reflection group of an Euclidean space if and only if $\Gamma$ is a finite Coxeter group. In this paper, we define {\it reflections} of geodesic spaces in general, and we prove that…

Let $G$ be a reductive group with Borel $B$ and Weyl group $W$. Then $B$-double cosets in $G$ are indexed by the Weyl group, say $O(w)$ for $w\in W$. Then we prove the minimal $B$-double coset in the convolution $O(w_1)*O(w_2)$ is…

This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems…

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take…

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 study actions of ``compact quantum groups'' on ``finite quantum spaces''. According to Woronowicz and to general $\c^*$-algebra philosophy these correspond to certain coactions $v:A\to A\otimes H$. Here $A$ is a finite dimensional…