Finite Coxeter Groups Research Articles

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.

On the structure constants of the descent algebra of the hyperoctahedral group

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulas involving Kronecker coefficients. Furthermore, using the theory of type B quasisymmetric functions introduced by Chow, we can extend our method to the hyperoctahedral group and derive some new formulas.

A divisibility result in combinatorics of generalized braids

For every finite Coxeter group Γ, each positive braid in the corresponding braid group admits a unique decomposition as a finite sequence of elements of Γ, the so-called Garside-normal form. The study of the associated adjacency matrix Adj(Γ) allows to count the number of Garside-normal form of a given length. In this paper we prove that the characteristic polynomial of Adj(Bn) divides the one of Adj(Bn+1). The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type A. We observe that this does not hold for type D. The other Coxeter types (I, E, F and H) are also studied.

ОБ АЛГОРИТМИЧЕСКИХ ПРОБЛЕМАХ В ГРУППАХ КОКСТЕРА

The main algorithmic problems of group theory posed by M. Dehn are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group’s isomorphism problem. Among the works related to the study of the M. Dehn’s problems, the most outstanding ones are the work of P. S. Novikov who proved the undecidability of the problem of words and the conjugacy problem for finitely presented groups as well as the undecidability of the problem of isomorphism of groups. In this regard, the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups were introduced by H. S. M. Coxeter: every reflection group is a Coxeter group if its generating elements are reflections with respect to hyperplanes limiting its fundamental polyhedron. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point. In an algebraic aspect Coxeter groups are studied starting with works by J. Tits who solved the problem of words in certain Coxeter groups. The article describes the known results obtained in solving algorithmic problems in Coxeter groups; the main purpose of the paper is to analyze of the results of solving algorithmic problems in Coxeter groups that were obtained by members of the Tula algebraic school ’Algorithmic problems of theory of the groups and semigroups ’ under the supervision of V. N. Bezverkhnii. It reviews assertions and theorems proved by the authors of the article for the various classes of Coxeter groups: Coxeter groups of large and extra-large types, Coxeter groups with a tree-structure, and Coxeter groups with n-angled structure. The basic approaches and methods of evidence among which the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii concerning the introduction of R-cancellations, special R-cancellations, special ring cancellations as well as method of graphs, method of types worked out by V. N. Bezverkhnii, method of special set of words designed by V. N. Bezverkhnii on the basis of the generalization of Nielsen method for free construction of groups. Classes of group considered in the article include all Coxeter groups which may be represented as generalized tree structures of Coxeter groups formed from Coxeter groups with tree structure with replacing some vertices of the corresponding tree-graph by Coxeter groups of large or extra-large types as well as Coxeter groups with n-angled structure.

The BMR freeness conjecture for the tetrahedral and octahedral families

ABSTRACTWe prove the validity of the freeness conjecture of Broué, Malle, and Rouquier for the generic Hecke algebras associated to the exceptional complex reflection groups of rank 2 belonging to the tetrahedral and octahedral families, and we give a description of the basis similar to the classical case of the finite Coxeter groups.

-GRAPHS AND GYOJA’S -GRAPH ALGEBRA

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

Mirror graphs: Graph theoretical characterization of reflection arrangements and finite Coxeter groups

Mirror graphs were introduced by Brešar et al. (2004) as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article we settle the structure of mirror graphs by characterizing them as precisely the Cayley graphs of the finite Coxeter groups or equivalently the tope graphs of reflection arrangements—well understood and classified structures. We provide a polynomial algorithm for their recognition.

The number of roots of full support

Chapoton has observed a simple product formula for the number of reflections in a finite Coxeter group that have full support. We give a uniform proof of his formula for Weyl groups. We furthermore refine his formula by the length of the roots.

The BMR freeness conjecture for the first two families of the exceptional groups of rank 2

The freeness conjecture of Broué, Malle and Rouquier for the generic Hecke algebras associated with complex reflection groups is still open for 14 cases, which cover almost all the exceptional complex reflection groups of rank 2. We prove this conjecture for 9 of these remaining cases, giving a basis similar to the classical case of the finite Coxeter groups.

Characterization of cyclically fully commutative elements in finite and affine Coxeter groups

An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. (2012). In particular the authors precisely identified the Coxeter groups having a finite number of cyclically fully commutative elements and enumerated them. In this work we characterize and enumerate those elements according to their Coxeter length in all finite and all affine Coxeter groups by using an operation on heaps, the cylindrical closure. In finite types, this refines the work of Boothby et al. (2012), by adding a new parameter. In affine type, all the results are new. In particular, we prove that there is a finite number of cyclically fully commutative logarithmic elements in all affine Coxeter groups. We also study the cyclically fully commutative involutions and prove that their number is finite in all Coxeter groups.

Vanishing ranges for the mod p cohomology of alternating subgroups of Coxeter groups

We obtain vanishing ranges for the mod p cohomology of alternating subgroups of finite p-free Coxeter groups. Here a Coxeter group W is p-free if the order of the product st is prime to p for every pair of Coxeter generators s,t of W. Our result generalizes those for alternating groups formerly proved by Kleshchev–Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite p-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxeter groups.

Involution words II: braid relations and atomic structures

Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted involutions $x$, $y$ in a Coxeter group $W$ with automorphism $*$, we associate a set of involution words $\hat{\mathcal{R}}_*(x,y)$. This set is the disjoint union of the reduced words of a set of group elements $\mathcal{A}_*(x,y)$, which we call the atoms of $y$ relative to $x$. The atoms, in turn, are contained in a larger set $\mathcal{B}_*(x,y) \subset W$ with a similar definition, whose elements we refer to as Hecke atoms. Our main results concern some interesting properties of the sets $\hat{\mathcal{R}}_*(x,y)$ and $\mathcal{A}_*(x,y) \subset \mathcal{B}_*(x,y)$. For finite Coxeter groups we prove that $\mathcal{A}_*(1,y)$ consists of exactly the minimal-length elements $w \in W$ such that $w^* y \leq w$ in Bruhat order, and conjecture a more general property for arbitrary Coxeter groups. In type $A$, we describe a simple set of conditions characterizing the sets $\mathcal{A}_*(x,y)$ for all involutions $x,y \in S_n$, giving a common generalization of three recent theorems of Can, Joyce, and Wyser. We show that the atoms of a fixed involution in the symmetric group (relative to $x=1$) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the "Chinese relation" studied by Cassaigne, Espie, et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of "braid relations" spanning the involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto's theorem for involution words in arbitrary Coxeter groups.

Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

Let (W,S) be a Coxeter system and write PW(q) for its Poincaré series. Lusztig has shown that the quotient PW(q2)/PW(q) is equal to a certain power series LW(q), defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in W. The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization LWJ(s,q)∈Z[[s,q]] depending on a subset J⊂S. This new power series specializes to LW(q) when s=−1 and is given explicitly by a sum of rational functions over the involutions which are minimal length representatives of the double cosets of the parabolic subgroup WJ in W. When W is an affine Weyl group, we consider the renormalized power series TW(s,q)=LWJ(s,q)/LW(q) with J given by the generating set of the corresponding finite Weyl group. We show that when W is an affine Weyl group of type A, the power series TW(s,q) is actually a polynomial in s and q with nonnegative coefficients, which turns out to be a q-analogue recently studied by Cigler of the Chebyshev polynomials of the first kind, arising in a completely different context.

On the Hurwitz action in finite Coxeter groups

Abstract We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

The goal of this paper is to compute the cuspidal Calogero–Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero–Moser space and then by classifying certain “rigid” modules. Numerical evidence suggests that there is a very close relationship between Calogero–Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero–Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.

Coxeter groups, quiver mutations and geometric manifolds

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action of a finite (or affine) Coxeter group. These CW-complexes undergo mutations agreeing with mutations of quivers and diagrams. We also generalize the construction to quivers and diagrams originating from unpunctured surfaces and orbifolds.

Multi-cluster complexes

We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.

Action of longest element on a Hecke algebra cell module

By a result of Mathas, the basis element Tw0 of the Hecke algebra of a finite Coxeter group acts in the canonical basis of a cell module as a permutation matrix times plus or minus a power of v. We generalize this result to the unequal parameter case. We also show that the image of Tw0 in the corresponding asymptotic Hecke algebra is given by a simple formula.

On involutions in symmetric groups and a conjecture of Lusztig

Let (W,S) be a Coxeter system equipped with a fixed automorphism ⁎ of order ≤2 which preserves S. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of “twisted” involutions (i.e., elements w∈W with w⁎=w−1) was naturally endowed with a module structure of the Hecke algebra of (W,S) with two distinguished bases, which can be viewed as twisted analogues of the well-known standard basis and Kazhdan–Lusztig basis. The transition matrix between these bases defines a family of polynomials Py,wσ which can be viewed as “twisted” analogues of the well-known Kazhdan–Lusztig polynomials of (W,S). Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter u2) associated to (W,S) generated by the element X∅:=∑w⁎=wu−ℓ(w)Tw. In this paper we prove this conjecture in the case when ⁎=id and W=Sn (the symmetric group on n letters). Our methods are expected to be generalised to all the other finite crystallographic Coxeter groups.

W-graph ideals and biideals

We further develop the theory of $$W$$ -graph ideals, first introduced in Howlett (J Algebra 361:188–212, 2012). We discuss $$W$$ -graph subideals, and induction and restriction of $$W$$ -graph ideals for parabolic subgroups. We introduce $$W$$ -graph biideals: those $$W$$ -graph ideals that yield $$(W\times W^{{\mathrm {o}}})$$ -graphs, where $$W^{{\mathrm {o}}}$$ is the group opposite to $$W$$ . We determine all $$W$$ -graph ideals and biideals in finite Coxeter groups of rank 2.