Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes

For any Coxeter system (W, S), the group W acts naturally on the complement of the associated complex hyperplane arrangement. By the well-known conjecture, the orbit space of this action is the classifying space of the corresponding Artin group. We describe some properties of configuration spaces of particles labeled by elements of a partial monoid and use them to prove that the orbit space mentioned in the conjecture is the classifying space of the positive Artin monoid. In particular, the conjecture reduces to a problem concerning the group completion of this monoid.     

Coxeter groups,Coxeter monoids and the Bruhat order

Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras—the q=0 case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.     

Sets of reflections defining twisted Bruhat orders

Twisted Bruhat orders are certain partial orders on a Coxeter system (W,S) associated to initial sections of reflection orders, which are certain subsets of the set of reflections T of a Coxeter system. We determine which subsets of T give rise to a partial order on W in the same way.     

A subalgebra of 0-Hecke algebra

Let (W,I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88] constructed a monoid structure on the set of all subsets of I using unipotent χ-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.     

Hecke algebras with independent parameters

We study the Hecke algebra \({\mathcal {H}}({\mathbf {q}})\) over an arbitrary field \({\mathbb {F}}\) of a Coxeter system (W, S) with independent parameters \({\mathbf {q}}=(q_s\in {\mathbb {F}}:s\in S)\) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of \({\mathcal {H}}({\mathbf {q}})\) could be as small as 1. We construct a basis for \({\mathcal {H}}({\mathbf {q}})\) when (W, S) is simply laced. We also characterize when \({\mathcal {H}}({\mathbf {q}})\) 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.     

A note on Weyl groups and root lattices

We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if (W, S) is a Coxeter system of finite rank n with set of reflections T and if \(t_1, \ldots t_n \in T\) are reflections in W that generate W, then \(P:= \langle t_1, \ldots t_{n-1}\rangle \) is a parabolic subgroup of (W, S) of rank \(n-1\) (Baumeister et al. in J Group Theory 20:103–131, 2017, Theorem 1.5). Here we show if (W, S) is crystallographic as well, then all the reflections \(t \in T\) such that \(\langle P, t\rangle = W\) form a single orbit under conjugation by P.     

Shortest path poset of Bruhat intervals

We define the shortest path poset SP(u,v) of a Bruhat interval [u,v], by considering the shortest u–v paths in the Bruhat graph of a Coxeter group W, where u,v∈W. We consider the case of SP(u,v) having a unique rising chain under a reflection order and show that in this case SP(u,v) is a Gorenstein^? poset. This allows us to derive the nonnegativity of certain coefficients of the complete cd-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition.     

The Hecke group algebra of a Coxeter group and its representation theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, …).In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well.This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases.     

Permutahedra and generalized associahedra

Given a finite Coxeter system (W,S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan Fc, 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.     

A quiver presentation for Solomon's descent algebra

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ(W) and an algorithm for the construction of a quiver presentation for the descent algebra Σ(W) of any given finite Coxeter group W.     

On the cyclically fully commutative elements of Coxeter groups

Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in?W. We say that w is cyclically fully commutative (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. Additionally, we characterize precisely which CFC elements have the property that powers of them remain fully commutative, via the presence of a simple combinatorial feature called a band. This allows us to give necessary and sufficient conditions for a CFC element w to be logarithmic, that is, ?(w ^k )=k??(w) for all k??1, for a large class of Coxeter groups that includes all affine Weyl groups and simply laced Coxeter groups. Finally, we give a simple non-CFC element that fails to be logarithmic under these conditions.     

Coxeter-biCatalan combinatorics

We pose counting problems related to the various settings for Coxeter-Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is to count “twin” pairs of objects from a corresponding problem in Coxeter-Catalan combinatorics. We show that the problems all have the same answer, and, for a given finite Coxeter group W, we call the common solution to these problems the W-biCatalan number. We compute the W-biCatalan number for all W and take the first steps in the study of Coxeter-biCatalan combinatorics.     

Admissible submonoids of Artin-Tits monoids

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.     

On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.     

Hecke-Kiselman monoids of small cardinality

In this paper, we give a characterization of digraphs Q, |Q|≤4 such that the associated Hecke-Kiselman monoids H _Q are finite. In general, a necessary condition for H _Q to be a finite monoid is that Q is acyclic and its Coxeter components are Dynkin diagrams. We show, by constructing examples, that such conditions are not sufficient.     

Cambrian lattices

For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a “cluster fan.” Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two “Tamari” lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.     

On the finite basis problem for certain 2-limited words

Let X* be a free monoid over an alphabet X and W be a finite language over X. Let S(W) be the Rees quotient X*/I(W), where I(W) is the ideal of X* consisting of all elements of X* that are not subwords of W. Then S(W) is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S(W) is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.     

A finiteness theorem for W-graphs

A W-graph for a Coxeter group W is a combinatorial structure that encodes a module for the group algebra of W, or more generally, a module for the associated Iwahori–Hecke algebra. Of special interest are the W-graphs that encode the action of the Hecke algebra on its Kazhdan–Lusztig basis, as well as the action on individual cells. In previous work, we isolated a few basic features common to the W-graphs in Kazhdan–Lusztig theory and used these to define the class of “admissible” W-graphs. The main result of this paper resolves one of the basic question about admissible W-graphs: there are only finitely many admissible W-cells (i.e., strongly connected admissible W-graphs) for each finite Coxeter group W. Ultimately, the finiteness depends only on the fact that admissible W-graphs have nonnegative integer edge weights. Indeed, we formulate a much more general finiteness theorem for “cells” in finite-dimensional algebras which in turn is fundamentally a finiteness theorem for nonnegative integer matrices satisfying a polynomial identity.