Related papers: Computing reflection length in an affine Coxeter g…

The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) is shown to be (q^n-1)^(n - 1). Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also…

If $\A$ is an algebra over a field $\F$ and $\s$ is a generating set of $\A$, the length of $\s$ indicates the maximal length needed to express an arbitrary element of $\A$ as a linear combination of words in the elements of $\s$. The…

We prove that affine Coxeter groups are profinitely rigid.

We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in $O(3)$. We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we…

We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is…

We study Soergel modules for arbitrary Coxeter groups. For infinite Coxeter groups, we show that the homomorphisms between Soergel modules are in general more than those coming from morphisms of Soergel bimodules. This result provides a…

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional…

Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t) = \sum_{x \in X} t^{\ell(x)}$, where $\ell$ is the length function on $W$. In this article we derive expressions for the…

In this paper we find some exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already…

A convex polytope $P$ in the real projective space with reflections in the facets of $P$ is a Coxeter polytope if the reflections generate a subgroup $\Gamma$ of the group of projective transformations so that the $\Gamma$-translates of the…

We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial…

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a…

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups $H_2$, $H_3$ and $H_4$. Using a…

We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras,…

We give several necessary and sufficient conditions for the existence of {\it the presentation by conjugation} for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence…

Let $P$ be a simplex in $S^n$ and $G_P$ be a group generated by the reflections with respect to the facets of $P$. We are interested in the case when the group $G_P$ is discrete. In this case we say that $G$ generates the discrete…

We describe the geometry of conjugation within any split subgroup $H$ of the full isometry group $G$ of $n$-dimensional Euclidean space. We prove that for any $h \in H$, the conjugacy class $[h]_H$ of $h$ is described geometrically by the…

We prove that the diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. This is achieved using Fourier analysis of unbounded measures on locally compact abelian groups as…

Let W be a Weyl group and let w be a Coxeter elememt of minimal length of W. In the early 1970's I.G.Macdonald stated that the trace of w on an irreducible representation of W is 0,1 or -1. In this paper we give a proof of this statement…

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group $W$, in terms of the spectrum of an associated operator,…