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In this paper we give a description of the structure of the based ring of the lowest two-sided cell for a weighted Coxeter group with complete graph. It is proved that this ring is generated, in a simple way, by some subrings which are isomorphic to the based rings of the lowest two-sided cells for affine Weyl groups of type A ˜ 1 or A ˜ 2 .

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$., Comment: 17 pages; errors corrected; example added. To appear in J. Symb. Comp

In this paper we prove that the parabolic Kazhdan–Lusztig polynomials and the parabolic R -polynomials of the universal Coxeter group can be computed in a combinatorial way, by using special matchings.

We study the concept of the Coxeter energy of graphs and digraphs (quivers) as an analogue of Gutman's adjacency energy, which has applications in theoretical chemistry and is a recently widely investigated graph invariant. Coxeter energy CE ( G ) of a (di)graph G is defined to be the sum of the absolute values of all complex eigenvalues of the Coxeter matrix associated with G. Our main inspiration for the study comes from the Coxeter formalism appearing in group theory, Lie theory, representation theory of algebras, mathematical physics and other contexts. We focus on the Coxeter energy of trees and we prove that the path (resp. the maximal star) has the smallest (resp. the greatest) Coxeter energy among all trees (resp. two large subclasses of trees) with fixed number of vertices. We provide several other related results, as the characterization of trees with second smallest and second greatest Coxeter energy, bounds for Coxeter energy, and general facts on Coxeter spectra of graphs extending known results e.g. for Salem trees and for certain special real Coxeter eigenvalues of trees. Additionally, we discuss few other energy-like quantities for the Coxeter spectra of (di)graphs, including Coxeter energy “normalized” by the trace of the Coxeter matrix and the quantities derived from variants of Coulson integral formula.

In well-known work, Kazhdan and Lusztig (1979) defined a new set of Hecke algebra basis elements (actually two such sets) associated to elements in any Coxeter group. Often these basis elements are computed by a standard recursive algorithm which, for Coxeter group elements of long length, generally involves computing most basis elements corresponding to Coxeter group elements of smaller length. Thus, many calculations simply compute all basis elements associated to a given length or less, even if the interest is in a specific Kazhdan-Lusztig basis element. Similar remarks apply to "parabolic" versions of these basis elements defined later by Deodhar (1987, 1990), though the lengths involved are the (smaller) lengths of distinguished coset representatives. We give an algorithm which targets any given Kazhdan-Lusztig basis element or parabolic analog and does not precompute any other Kazhdan-Lusztig basis elements. In particular it does not have to store them. This results in a considerable saving in memory usage, enabling new calculations in an important case (for finite and algebraic group 1-cohomology with irreducible coefficients) analyzed by Scott-Xi (2010).

We introduce and study in detail so-called circulant (Coxeter-periodic) elements and circulant families in a bilinear lattice K as well as their dual versions, called anti-circulant. We show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the Coxeter polynomial χ K of K and in consequence, of Coxeter polynomials of algebras of finite global dimension. We discuss the properties of quadratic forms induced by circulant and anti-circulant families. Moreover, we interpret the results in the language of representation theory of algebras and point out applications (facts concerning tubular families in Auslander–Reiten quivers and quadratic forms of algebras). Abstract considerations in bilinear lattices are illustrated with a collection of non-trivial examples arising from module and derived categories. The results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts.

Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$ of the polynomial ring $\mathbb{C}[x_1, \dots, x_n]$ in $n$ variables. When $k = n$, both of these quotients coincide with the classical coinvariant algebra attached to $G_n$. The algebraic properties of our quotients are governed by the combinatorial properties of $k$-dimensional faces in the Coxeter complex attached to $G_n$ (in the case of $R_{n,k}$) and $r$-colored ordered set partitions of $\{1, 2, \dots, n\}$ with $k$ blocks (in the case of $S_{n,k}$). Our work generalizes a construction of Haglund, Rhoades, and Shimozono from the symmetric group $\mathfrak{S}_n$ to the more general wreath products $G_n$., Comment: 44 pages, typos fixed and notation clarified

The Hecke algebra $\mathbb{C}_q[W]$ of a Coxter group $W$, associated to parameter $q$, can be completed to a von Neumann algebra $\mathcal{N}_q(W)$. We study such algebras in case where $W$ is right-angled. We determine the range of $q$ for which $\mathcal{N}_q(W)$ is a factor, i.e. has trivial center. Moreover, in case of nontrivial center, we prove a result allowing to decompose $\mathcal{N}_q(W)$ into a finite direct sum of factors., Comment: 15 pages, 5 figures

We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxeter graph. Supporting this conjecture, we prove that every planar graph of odd-girth at least 17 admits a homomorphism to the Coxeter graph.

The affine Weyl group ( C ˜ n , S ) can be realized as the fixed point set of the affine Weyl group ( A ˜ m , S ˜ m ) , m ∈ { 2 n − 1 , 2 n , 2 n + 1 } , under a certain group automorphism α m , n . Let l ˜ m be the length function of A ˜ m . The aim of the present paper is to give a combinatorial description for all the left cells of A ˜ m which have non-empty intersection with C ˜ n . Then we use this description to deduce some formulae for the number of left cells of the weighted Coxeter group ( C ˜ n , l ˜ m ) in the set E λ associated to any partition λ of m + 1 .

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type $A$, we connect our results to Fan--Green's cell structure of the corresponding Temperley--Lieb algebra., 25 pages

Lusztig proposed 15 conjectures for any weighted Coxeter groups. In this study, we explicitly describe Kazhdan–Lusztig cells, distinguished involutions and the function a for any weighted universal Coxeter group. We also verify these conjectures for this group.

We show that every finitely generated Artin–Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.

Inspired by famous results of Drozd (1974) [13], Nazarova–Roĭter (1975) [42], and Bondarenko–Stepochkina (2005) [8], on the importance of finite posets I with the non-negative Tits quadratic form qˆI:ZI→Z in the representation theory, we study the finite non-negative posets I of corank two, that is, such posets I that the symmetric Gram matrix GI:=12[CI+CItr]∈MI(Q) of I is positive semi-definite of corank two, where CI∈MI(Z) is the incidence matrix of I. A structure of such posets is described in the paper. It is shown that every such a poset I can be obtained in a natural way from a corank zero poset J and a pair of roots of J. Following main ideas of the Coxeter spectral analysis of posets, we associate with any connected corank two poset I the incidence quadratic form qI:ZI→Z, the Coxeter matrix CoxI:=−CI⋅CI−tr, the Coxeter polynomial coxI(t):=det⁡(t⋅E−CoxI)∈Z[t], its complex Coxeter spectrum speccI, the Coxeter number cI∈N∪{∞}, the reduced Coxeter number cˇI∈N, the incidence defect homomorphism ∂˜I:ZI→KerqI⊆ZI, and a simply laced Dynkin diagram DynI. In particular, we study the question when the Coxeter spectrum speccI of I, together with DynI, determines the incidence matrix CI of I (hence the poset I) uniquely, up to a Z-congruence. By applying linear algebra techniques and a combinatorial computer-aided analysis of posets, we prove that the question has an affirmative answer for connected corank two posets I, with at most 15 elements. A complete list of such posets is presented and their Coxeter type invariants are computed. It follows that |I|≥6, there is only one such a poset I of cardinality 6, and 8 such posets of cardinality 7, up to the poset duality I↦Iop. The existence of a ΦI-mesh translation quiver Γ(RI∪KerqI,ΦI) on the set RI∪KerqI is discussed, where RI={v∈ZI;qI(v)=1} is the set of roots of qI.

Let ( W , S ) be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red ( z ) of all reduced expressions for any z ∈ W . By associating each bc-expression to a certain symbol, we describe the set Red ( z ) and compute its cardinal | Red ( z ) | in terms of symbols. An explicit formula for | Red ( z ) | is deduced, where the Fibonacci numbers play a crucial role.

Let Λ be a k-algebra of finite global dimension. We study tubular families in the Auslander–Reiten quiver of the bounded derived category Db(Λ) satisfying certain natural axioms. In particular, we precisely describe their influence on the cyclotomic factors of the Coxeter polynomial χΛ of Λ and discuss several numerical limitations for their possible shapes. Moreover, we show that our results provide an alternative, relatively simple proof of non-trivial classical facts concerning tubular families in module categories, and also extend them slightly.

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in W_0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w in W for which $\ell(w^n)=n\ell(w)$ for any integer n. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight., 12 pages, to appear in Journal of Algebra

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan–Lusztig cells in infinite Coxeter groups W. Our main focus is the set D of distinguished involutions in W, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set D has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of D we assign an explicitly defined set of equivalence relations on W that altogether conjecturally determine the partition of W into left (right) cells. We are able to prove these conjectures only in a special case, but even from these partial results we can deduce some interesting corollaries.

Let W be a Coxeter group and WJ a finite parabolic subgroup. We present here a geometric new proof of X. Heʼs result on minimal length elements of an orbit of WJ which acts on W in a certain way.

We present a complete list of the Coxeter polynomials of P-critical bigraphs Δ, for | Δ 0 | ⩽ 8 . Moreover, given P-critical bigraphs Δ, Δ ′ of the Euclidean type D Δ = A ˜ n − 1 and n ⩽ 8 , we establish by a computer calculation the equivalence of the following three statements: (a) cox Δ ( t ) = cox Δ ′ ( t ) , (b) the reduced Coxeter numbers c Δ and c Δ ′ of Δ and Δ ′ coincide, (c) the tubular types tub Δ and tub Δ ′ of Δ and Δ ′ coincide (see Section 2).

Consider an irreducible finite Coxeter system. We show that for any nonnegative integer n the sum of the nth powers of the Coxeter exponents can be written uniformly as a polynomial in four parameters: h (the Coxeter number), r (the rank), and two further parameters., Comment: 14 pages

If x and y are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group W, we say that x dominates y if for all w ∈ W , wy is a negative root whenever wx is a negative root. We call a positive root elementary if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by E . It has been proved by B. Brink and R.B. Howlett [B. Brink, R.B. Howlett, A finiteness property and an automatic structure of Coxeter groups, Math. Ann. 296 (1993) 179–190] that E is finite if (and only if) W is a finite-rank Coxeter group. Amongst other things, this finiteness property enabled Brink and Howlett to establish the automaticity of all finite-rank Coxeter groups. Later Brink has also given a complete description of the set E for arbitrary finite-rank Coxeter groups [B. Brink, The set of dominance-minimal roots, J. Algebra 206 (1998) 371–412]. However the set of non-elementary positive roots has received little attention in the literature. In this paper we answer a collection of questions concerning the dominance behavior between such non-elementary positive roots. In particular, we show that for any finite-rank Coxeter group and for any non-negative integer n, the set of roots each dominating precisely n other positive roots is finite. We give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation.

In "W-graph ideals" (Robert B. Howlett and Van Minh Nguyen) the concept of a W-graph ideal in a Coxeter group was introduced, and it was shown how a W-graph can be constructed from a given W-graph ideal. In this paper, we describe a class of W-graph ideals from which certain Kazhdan-Lusztig left cells arise. The result justifies the algorithm as illustrated in "W-graph ideals" for the construction of W-graphs for Specht modules for the Hecke algebra of type A., Comment: 16 pages

We present a new axiomatization of what is known as the “root system” of a Coxeter group that does not involve vector spaces or Coxeter groups. We use this to combinatorially characterize the structure of intervals in weak Bruhat order in much the same way that finite distributive lattices are characterized as the lattices of order ideals of partially ordered sets. In fact, the result for distributive lattices follows as a special case.

We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian statistic, which we call the sorting index. The sorting index of a permutation and its type B and type D analogues have natural combinatorial descriptions which we describe in detail., 14 pages, minor changes, new references added

Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the cones in the $c$-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if $W$ is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to $W$, then we show that our realizations have integer coordinates in this lattice., Comment: 27 pages, 10 figures; v3: 31 pages, 10 figures, Section 3 is rewritten, corrected typos, and updated references

Soergel bimodule category B is a categorification of the Hecke algebra of a Coxeter system (W,S). We find a presentation of B (as a tensor category) by generators and relations when W is a right-angled Coxeter group., 34 pages

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms b ˆ I , b I , b ¯ I , b I • : Z I × Z I → Z (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z -bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form q ¯ I ( x ) = b ¯ I ( x , x ) weakly positive (resp. with the reduced Euler form q I • ( x ) = b I • ( x , x ) weakly positive), and posets I, with the Tits form q ˆ I ( x ) = b ¯ I ( x , x ) weakly positive.

We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincare series. This uses Coxeter elements involving Eichler–Siegel transformations. We give geometrical interpretations for the lattices and isometries involved, lifting them to triangulated categories.

In 1939 Coxeter published three infinite families of group presentations. He studied their properties, in particular determining when groups defined by members of the families are infinite and the structure of finite ones. Eight presentations remained for which the finiteness question was unsettled. We show that two of these eight presentations define finite groups (for which we give comprehensive proofs and provide detailed structural information) and that two of the presentations define infinite groups. Our results rely on substantial amounts of computer calculations, in particular on coset enumeration to prove finiteness and on computation of automatic structures using Knuth–Bendix rewriting to prove infiniteness.

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups.

This paper deals with infinite Coxeter groups. We use geometric techniques to prove two main results. One is of Lie-theoretic nature; it shows the existence of many hyperbolic configurations of three pairwise disjoint roots in a given Coxeter complex, provided it is not an Euclidean tiling. The other is both of algebraic and measure-theoretic nature since it deals with Hecke algebras; it shows that for automorphism groups of buildings, convolution algebras of bi-invariant functions are never commutative, provided the building is not Euclidean. Proofs are of geometric nature: the main idea is to exhibit and use enough trees of valency ⩾3 inside a non-affine Coxeter complex.

In this paper we show that for a simply-laced root system a choice of a Coxeter element C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset Iˆ of I×Z2h, where h is the Coxeter number. The set Iˆ has a natural quiver structure; we call it the periodic Auslander–Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of Iˆ, and the root lattice and the inner product admit an explicit description in terms of Iˆ. Finally, we relate this construction to the theory of quiver representations.

Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering CD(A) by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W., Comment: Final version, 25 pages, to appear in the Journal of Algebra. Minor word changes. A new section (6) has been added where the rigidity of quasi-Coxeter algebra structures on the group algebra of a finite Coxeter group is proved

Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $\chi$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type., Comment: 12 pages, to appear in J. Algebra

We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat–Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm–Liouville equations.

This paper gives examples of embeddings of root systems of Coxeter groups, including sporadic embeddings of standard real root systems in other standard real root systems, and, for a general Coxeter group, an embedding of its universal symmetric root system over commutative rings into the standard real root system of a simply laced Coxeter group.

In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $\Delta_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that $\Delta_T$ is $(n-b-1)$-connected when the tree has $b$ leaves. We provide a shelling for the $(n-b)$-skeleton of $\Delta_T$, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree $T$ which imply shellability of $\Delta_T$, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes $M_{m,n}$ with $n \ge 2m-1$. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one., Comment: 23 pages

In this paper we show that the cyclotomic factors of the E n Coxeter polynomials depend only on the value of n mod 360 , and come exclusively from spherical subdiagrams.

The concept of W-graph was introduced in the influential paper [David Kazhdan, George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] of Kazhdan and Lusztig. If the underlying Coxeter group is the symmetric group, then Kazhdan and Lusztig showed that every irreducible representation arises from a left cell and, hence, is given by a W-graph. This is the optimal picture that can hope for. For other types of Coxeter groups, the representations arising from left cells are no longer irreducible. However, Gyoja [A. Gyoja, On the existence of a W-graph for an irreducible representation of a Coxeter group, J. Algebra 86 (1984) 422–438. [3] ] proved, by a general argument, that every irreducible representation of a Hecke algebra associated with a finite Coxeter group is given by a W-graph, but this is a pure existence result, and the question remained open of how to construct such W-graphs explicitly. In [R.B. Howlett, Yunchuan Yin, Inducing W-graphs, Math. Z. 244 (2003) 415–431] we provided a general method for producing W-graphs, by induction of W-graphs from parabolic subgroups, and then [Yunchuan Yin, Irreducible W-graphs for type D 4 and D 5 , Comm. Algebra 34 (2006) 547–565. [7] ] we constructed all the irreducible W-graphs for type D 4 and D 5 by hand calculation. In this paper we will show that the algorithm introduced in [R.B. Howlett, Yunchuan Yin, Inducing W-graphs, Math. Z. 244 (2003) 415–431] is sufficiently powerful to construct explicit W-graphs for all irreducible representations of Hecke algebras of type E 6 and E 7 . The computer algebra system “Magma” was used for the calculations.

Let Λ be a finite-dimensional algebra over an algebraically closed field k of finite global dimension. Let M be a finitely generated Λ-module and let Γ=Λ[M] be the one point extension algebra. We show how to compute the Coxeter polynomial for Γ from the Coxeter polynomial of Λ and homological invariants of M.

A new proof is given for the statement: For an irreducible, infinite Coxeter group ( W , S ) and w ∈ W , if wSw - 1 = S , then w = 1 (the identity element of W).

In [R.P. Stanley, The descent set and connectivity set of a permutation, J. Integer Seq. 8 (3) (2005) Article 05.3.8] Stanley gives certain enumerative identities revealing a duality between descent sets and connectivity sets of the symmetric group. In this paper we generalize these identities to all Coxeter groups. The proofs are obtained by giving these identities an algebraic explanation in terms of parabolic subgroups, coset representatives, and Poincaré series, and by a formal argument in terms of inclusion–exclusion-like matrices.

Abstract Young (briefly: AY) representations are Coxeter group representations which carry an extended Young form. These representations appear in a new axiomatic approach to the representation theory of Coxeter groups. While AY representations for Coxeter groups of types A and B are well understood, little is known about type D. We introduce D-Young tableaux and study minimal AY representations associated with them. The main result is a decomposition rule for the AY representations which are induced from Sn into Dn. The proof of this rule involves a combinatorial bijection together with theorems from complex analysis.

We show the analogue of Mühlherr’s [B. Mühlherr, Coxeter groups in Coxeter groups, in: Finite Geometry and Combinatorics, Cambridge University Press, 1993, pp. 277–287] for Artin–Tits monoids and for Artin–Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin–Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin–Tits monoid (resp. group).This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin–Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the notion of LCM-homomorphisms defined by Crisp in [J. Crisp, Injective maps between Artin groups, in: Geom. Group Theory Down Under (Canberra 1996), de Gruyter, Berlin, 1999, pp. 119–137] and generalized by Godelle in [E. Godelle, Morphismes injectifs entre groupes d’Artin-Tits, Algebr. Geom. Topol. 2 (2002) 519–536].We then complete the classification of the admissible partitions for which the Coxeter graphs involved have no infinite label, started by Mühlherr in [B. Mühlherr, Some contributions to the theory of buildings based on the gate property, Dissertation, Tübingen, 1994]. This leads us to the classification of Crisp’s LCM-homomorphisms.

Coxeter cones are formed by intersecting the nonnegative sides of a collection of root hyperplanes in some root system. They are shellable subcomplexes of the Coxeter complex, and their h-vectors record the distribution of descents among their chambers. We identify a natural class of “graded” Coxeter cones with the property that their h-vectors are symmetric and unimodal, thereby generalizing recent theorems of Reiner–Welker and Brändén about the Eulerian polynomials of graded partially ordered sets.

We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property., Comment: 9 pages

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 Weyl group of type G ˜ 2 . Using explicit computation with COXETER and CHEVIE , we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. A key ingredient of the proof is a general result which shows that the Kazhdan–Lusztig polynomials of affine Weyl group are invariant under (large enough) translations.

For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ -spaces with isotropy on the family V C of virtually cyclic subgroups of Γ . We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O + ( n , 1 ) = Isom ( H n ) . We use the information we obtain to explicitly compute the lower algebraic K -theory of the Coxeter group Γ 3 (a non-uniform lattice in O + ( 3 , 1 ) ). Part of this computation involves calculating certain Waldhausen Nil-groups for Z [ D 2 ] , Z [ D 3 ] .

We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.