Coxeter Diagram Research Articles

Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra \mathbb{C}[\Gamma] with \Gamma a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten \mathcal{S}_{p} class p \in [2, \infty) , then the n -fold tensor power H^{\otimes n}_{\Gamma} for n \geq \frac{p}{2} is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in \mathcal{S}_{p} for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient- \mathcal{S}_{p} property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p -integrable representation.

City products of right-angled buildings and their universal groups

We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones. More precisely, if M is a right-angled Coxeter diagram of rank n and Δ1,…,Δn are right-angled buildings, then we construct a new right-angled building ▪. We can recover the buildings Δ1,…,Δn as residues of Δ, but we can also construct a skeletal building of type M from Δ that captures the large-scale geometry of Δ.We then proceed to study universal groups for city products of right-angled buildings, and we show that the universal group of Δ can be expressed in terms of the universal groups for the buildings Δ1,…,Δn and the structure of M. As an application, we show the existence of many examples of pairs of different buildings of the same type that admit (topologically) isomorphic universal groups, thereby vastly generalizing a recent example by Lara Beßmann.

Proper locally spherical hypertopes of hyperbolic type

Given any irreducible Coxeter group C of hyperbolic type with nonlinear diagram and rank at least 4, whose maximal parabolic subgroups are finite, we construct an infinite family of locally spherical regular hypertopes of hyperbolic type whose Coxeter diagram is the same as that of C.

Stability of Stretched Root Systems, Root Posets and Shards

Inspired by the infinite families of finite and affine root systems, we define a "stretching" operation on general crystallographic root systems which, on the level of Coxeter diagrams, replaces a vertex with a path of unlabeled edges. We embed a root system into its stretched versions using a similar operation on individual roots. For a fixed root, we describe the long-term behavior of two associated structures as we lengthen the stretched path: the downset in the root poset and Reading's arrangement of shards. We show that both eventually admit a uniform description, and deduce enumerative consequences: the size of the downset is eventually a polynomial, and the number of shards grows exponentially.

Constructing highly regular expanders from hyperbolic Coxeter groups

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders globally and locally. They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled.

The low-dimensional homology of finite-rank Coxeter groups

We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.

Double Affine Hecke Algebras and Congruence Groups

The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type $A_{2}$, which descends to a faithful outer action of a congruence subgroup of $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$. This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. The structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products $W\ltimes Q^{\vee}$ of the Weyl group $W$ with the coroot lattice $Q^{\vee}$. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type $A_{2},B_{2},G_{2}$ act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist $r=1,2,3$, respectively. This extends a fundamental result of Cherednik for $r=1$. We show further that the above rank two Artin group action descends to an outer action of congruence subgroup $\Gamma_{1}(r)$. In particular $\Gamma_{1}( r) $ acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist type $r$, giving rise to a projective representation of $\Gamma_{1}( r) $ on the space of a $\Gamma_{1}( r) $-stable representation.

Locally spherical hypertopes from generalised cubes

We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope. These hypertopes are related to the semi-regular polyotopes with a tail-triangle Coxeter diagram constructed by Monson and Schulte. We discuss several interesting examples derived when this construction is applied to generalised cubes. In particular, we produce an example of a rank 5 finite locally spherical proper hypertope of hyperbolic type. No such examples were previously known.

A taxonomy of crystallographic sphere packings

This paper seeks to catalogue and examine crystallographic sphere packings as defined by Kontorovich and Nakamura. There exist at least three sources which give rise to crystallographic packings, namely polyhedra, reflective extended Bianchi groups, and various higher dimensional quadratic forms. When applied in conjunction with the Koebe–Andreev–Thurston Theorem, Kontorovich and Nakamura's Structure Theorem guarantees crystallographic packings to be generated from polyhedra in n=2. The Structure Theorem similarly allows us to generate packings from the reflective extended Bianchi groups in n=2 by applying Vinberg's algorithm to obtain the appropriate Coxeter diagrams. In n>2, the Structure Theorem when used with Vinberg's algorithm allows us to explore whether certain Coxeter diagrams in Hn+1 for a given quadratic form admit a packing at all. Kontorovich and Nakamura's Finiteness Theorem shows that there exist only finitely many classes of superintegral such packings, all of which exist in dimensions n⩽20. In this work, we systematically determine all known examples of crystallographic sphere packings.

Tits Triangles

AbstractA Tits polygon is a bipartite graph in which the neighborhood of every vertex is endowed with an “opposition relation” satisfying certain properties. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. There is a standard construction that produces a Tits polygon whose opposition relations are not all trivial from an arbitrary pair $(\unicode[STIX]{x1D6E5},T)$, where $\unicode[STIX]{x1D6E5}$ is a building of type $\unicode[STIX]{x1D6F1}$, $\unicode[STIX]{x1D6F1}$ is a spherical, irreducible Coxeter diagram of rank at least $3$, and $T$ is a Tits index of absolute type $\unicode[STIX]{x1D6F1}$ and relative rank $2$. A Tits polygon is called $k$-plump if its opposition relations satisfy a mild condition that is satisfied by all Tits triangles coming from a pair $(\unicode[STIX]{x1D6E5},T)$ such that every panel of $\unicode[STIX]{x1D6E5}$ has at least $k+1$ chambers. We show that a $5$-plump Tits triangle is parametrized and uniquely determined by a ring $R$ that is alternative and of stable rank $2$. We use the connection between Tits triangles and the theory of Veldkamp planes as developed by Veldkamp and Faulkner to show existence.

On the Subregular J-Rings of Coxeter Systems

We recall Lusztig's construction of the asymptotic Hecke algebra $J$ of a Coxeter system $(W,S)$ via the Kazhdan--Lusztig basis of the corresponding Hecke algebra. The algebra $J$ has a direct summand $J_E$ for each two-sided Kazhdan--Lusztig cell of $W$, and we study the summand $J_C$ corresponding to a particular cell $C$ called the subregular cell. We develop a combinatorial method to compute $J_C$ without using the Kazhdan--Lusztig basis. As applications, we deduce some connections between $J_C$ and the Coxeter diagram of $W$, and we show that for certain Coxeter systems $J_C$ contains subalgebras that are free fusion rings in the sense of [Banica], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.

Elementary equivalence in Artin groups of finite type

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The [Formula: see text] fragment suffices to distinguish the elementary theories of the groups in question. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces — a geometric analogue of braid groups — are elementarily equivalent if and only if they are isomorphic.

Walls in Milnor Fiber Complexes

For a real reflection group the reflecting hyperplanes cut out on the unit sphere a simplicial complex called the Coxeter complex. Abramenko showed that each reflecting hyperplane meets the Coxeter complex in another Coxeter complex if and only if the Coxeter diagram contains no subdiagram of type D_4 , F_4 or H_4 . The present paper extends Abramenko's result to a wider class of complex reflection groups. These groups have a Coxeter-like presentation and a Coxeter-like complex called the Milnor fiber complex. Our first main theorem classifies the groups whose reflecting hyperplanes meet the Milnor fiber complex in another Milnor fiber complex. To understand better the walls that fail to be Milnor fiber complexes we introduce Milnor walls. Our second main theorem generalizes Abramenko's result in a second way. It says that each wall of a Milnor fiber complex is a Milnor wall if and only if the diagram contains no subdiagram of type D_4 , F_4 , or H_4 .

Hyperbolic geometry and moduli of real curves of genus three

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to maximal real quartic curves, we obtain a more explicit description. We construct a Coxeter diagram that encodes the geometry of this component.

Hecke algebras with independent parameters

We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group $W$. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of $\H(\bq)$ could be as small as $1$. We construct a basis for $\H(\bq)$ when $(W,S)$ is simply laced. We also characterize when $\H(\bq)$ is commutative, which happens only if the Coxeter diagram of $(W,S)$ is simply laced and bipartite. In particular, for type A we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements

In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_{\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\gamma\in W$ the lattice $\mathcal{B}_{\gamma}$ is in fact distributive. It turns out that this is the case for the "coincidental" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\gamma$ of said groups for which $\mathcal{B}_{\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.

Projection of Polyhedra onto Coxeter Planes Described with Quaternions

3-dimensional convex uniform polyhedra have been projected onto their corresponding Coxeter planes defined by the simple roots of the Coxeter diagram ,

Subword complexes and edge subdivisions

For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair (Q, π), where Q is a word in the alphabet of simple reflections and π is a group element. We discuss the transformations of such a complex that are induced by braid moves of the word Q. We show that under certain conditions, such a transformation is a composition of edge subdivisions and inverse edge subdivisions. In this case, we describe how the H- and γ-polynomials change under the transformation. This case includes all braid moves for groups with simply laced Coxeter diagrams.

Embeddings of local fields in simple algebras and simplicial structures

We give a geometric interpretation of Broussous{Grabitz embedding types. We fix a central division algebra D of finite index over a non-Archimedean local field F and a positive integer m. Further we fix a hereditary order a of Mm(D) and an unramified field extension EjF in Mm(D) which is embeddable in D and which normalizes a. Such a pair (E, a) is called an embedding. The embedding types classify the GLm(D)-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of (E, a) from the simplicial type of the image of the barycenter of a under the canonical isomorphism, from the set of Ex-fixed points of the reduced building of GLm(D) to the reduced building of the centralizer of Ex in GLm(D). Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.

Reflection centralizers in Coxeter groups

We refine Brink’s theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for the centralizer, which is finitely generated when W is. And we give a method for computing the Coxeter diagram for its reflection subgroup. In many cases, our method allows one to compute centralizers in one’s head.