Related papers: Three-way tiling sets in two dimensions
Random walk on the chambers of hyperplanes arrangements is used to define a family of card shuffling measures $H_{W,x}$ for a finite Coxeter group W and real $x \neq 0$. By algebraic group theory, there is a map from the semisimple orbits…
We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda$ in such a way that each point of the space gets covered exactly $k$ times,…
In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling.
In [5], Elnitsky constructed three elegant bijections between classes of reduced words for Type $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{D}$ families of Coxeter groups and certain tilings of polygons. This paper offers a particular…
Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of…
There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration…
Let (W,S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits' bilinear form associated to the standard root system of…
We find and study a six-parameter family of coupled Painlev\'e III systems in dimension six with affine Weyl group symmetry of type $D_6^{(1)}$. We also find and study its degenerate systems with affine Weyl group symmetry of types…
Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine…
A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface…
In the study of walks with small steps confined to multidimensional orthants, a certain group of transformations plays a central role. In particular, several techniques to potentially compute the generating function, including the orbit sum…
A Gelfand-Tsetlin scheme of depth N is a triangular array with m integers at level m, m=1,...,N, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed N-th row. We…
Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains…
Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities $\leq 2$, where for nullity 2 there is only one known such presentation. We…
In this note, we characterize affine and non-affine Coxeter systems among all Coxeter systems in terms of the structure of their reflection orders. For an infinite irreducible system $(W,S)$, we show that affineness can be characterized in…
We use the author's combinatorial theory of full heaps (defined in math.QA/0605768) to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of…
In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of…
The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible…
By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction…
This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all…