Coxeter System Research Articles

Reduced expression of minimal infinite reduced words of affine Weyl groups

For an infinite Coxeter system, one can extend the weak right order to the set of equivalence classes of infinite reduced words. This is called limit weak order. In Lam and Pylyavskyy (2013) [12], Lam and Pylyavskyy showed that for affine Weyl groups of type A˜n, minimal elements of the limit weak order are equivalent to infinite Coxeter elements and asked the question of characterization, in terms of infinite reduced words, of the minimal elements of the limit weak order for other affine types. In this paper we give such a characterization by determining the reduced expression of a representative infinite reduced word for each of these minimal elements.

Parametrization, structure and Bruhat order of certain spherical quotients

Let G G be a reductive algebraic group and let Z Z be the stabilizer of a nilpotent element e e of the Lie algebra of G G . We consider the action of Z Z on the flag variety of G G , and we focus on the case where this action has a finite number of orbits (i.e., Z Z is a spherical subgroup). This holds for instance if e e has height 2 2 . In this case we give a parametrization of the Z Z -orbits and we show that each Z Z -orbit has a structure of algebraic affine bundle. In particular, in type A A , we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type A A , we show that the Bruhat order of the Z Z -orbits can be described in this way.

A bimodule description of the Hecke category

AbstractFor a Coxeter system and a representation$V$of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when$V$is reflection faithful. Elias and Williamson defined another category when$V$is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when$V$is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on$V$. In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.

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.

N-low elements and maximal rank k reflection subgroups of Coxeter groups

This paper proves the conjectures (previously proved in the case n=0) that for any Coxeter system (W,S) and any natural number n, the set of n-small roots for W is bipodal and the set of n-low elements of W is a Garside shadow. The proof here uses special cases (k=3) of properties (notably, their existence) of certain maximal rank k reflection subgroups of W, for non-negative integers k.

Reflection trees of graphs as boundaries of Coxeter groups

To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $\Gamma$ for which $X$ is the geometric realization) then the visual boundary $\partial_\infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $\Gamma$ is homeomorphic to ${\cal X}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\cal X}^r(X)$.

Automorphisms of odd Coxeter groups

An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $\mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups. As applications, we prove that Coxeter groups in this family satisfy the $R_\infty$-property and are (co)-Hopfian. We compare structural properties, automorphism groups, $\R_\infty$-property and (co)-Hopfianity of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group.

Locally finite continuations and Coxeter groups of infinite ranks

An involution r in a Coxeter group W is called an intrinsic reflection of W if r∈SW for each Coxeter generating set S of W. In recent joint work with R.B. Howlett [13] we determined all intrinsic reflections in finitely generated Coxeter groups. In the present paper we extend this result to the infinite rank case. An important tool in [13] is the notion of the finite continuation of an involution that is only meaningful for finitely generated Coxeter groups. Here we introduce the locally finite continuation for any subset of an arbitrary group which enables us to deal with Coxeter groups of infinite rank. We apply our result to show that certain classes of Coxeter groups are reflection independent and we investigate rigidity of 2-spherical Coxeter systems of arbitrary ranks.

Oriented matroid structures from realized root systems

This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter system in real vector spaces. We settle the question for finite Coxeter systems, irreducible affine Weyl groups and all rank three Coxeter systems. In these cases, the oriented matroid structure is unique unless W is of type $${\widetilde{A}}_n, n\ge 3$$ , in which case there are three possibilities.

Cataland: Why the Fuss?

The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncross- ing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss–Catalan generalizations. We provide new viewpoints for these, introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.

A two-sided analogue of the Coxeter complex

For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . There is exactly one maximal face for each element of the group W . The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the h-polynomial is given by the “two-sided” W -Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in W .

Parabolic double cosets in Coxeter groups

Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.

The ABC of p-cells

Parallel to the very rich theory of Kazhdan–Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson in Categorification and higher representation theory, American Mathematical Society, Providence, 2017). Our main technical tools are the star-operations introduced by Kazhdan–Lusztig (Invent. Math. 53(2):65–184, 1979) which have interesting numerical consequences for the p-canonical basis. As an application, we explicitly describe p-cells in finite type A (i.e. for symmetric groups) using the Robinson–Schensted correspondence. Moreover, we show that Kazhdan–Lusztig cells in finite types B and C decompose into p-cells for $$p > 2$$ p > 2 .

Generalized Burnside algebra of type B_{n}

In this paper, we firstly give an alternative method to determine the size of $C(S_{n})$ which is the set of elements of type $S_{n}$ in a finite Coxeter system $(W_{n},S_{n})$ of type $B_{n}$. We also show that all cuspidal classes of $W_{n}$ are actually the conjugate classes $\mathcal{K}_{\lambda}$ for every $\lambda \in \mathcal{DP}^{+}(n)$. We then define the generalized Burnside algebra $HB(W_{n})$ for $W_{n}$ and construct a surjective algebra morphism between $HB(W_{n})$ and Mantaci-Reutenauer algebra $\mathcal{MR}(W_{n})$. We obtain a set of orthogonal primitive idempotents $e_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $HB(W_{n})$, that is, all the characteristic class functions of $W_{n}$. Finally, we give an effective formula to compute the number of elements of all the conjugate classes $\mathcal{K}_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $W_{n}$.

Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras

For a right-angled Coxeter system $(W,S)$ and $q>0$, let $\mathcal{M}_q$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators $T_s, s \in S$ satisfying the Hecke relation $(\sqrt{q}\: T_s - q) (\sqrt{q} \: T_s + 1) = 0$ as well as suitable commutation relations. Under the assumption that $(W, S)$ is irreducible and $\vert S \vert \geq 3$ it was proved by Garncarek that $\mathcal{M}_q$ is a factor (of type II$_1$) for a range $q \in [\rho, \rho^{-1}]$ and otherwise $\mathcal{M}_q$ is the direct sum of a II$_1$-factor and $\mathbb{C}$. In this paper we prove (under the same natural conditions as Garncarek) that $\mathcal{M}_q$ is non-injective, that it has the weak-$\ast$ completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case $\mathcal{M}_q$ is a strongly solid algebra and consequently $\mathcal{M}_q$ cannot have a Cartan subalgebra. In the general case $\mathcal{M}_q$ need not be strongly solid. However, we give examples of non-hyperbolic right-angled Coxeter groups such that $\mathcal{M}_q$ does not possess a Cartan subalgebra.

Hecke algebras of simply-laced type with independent parameters

We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. Our results demonstrate an interesting combination of the representation theory of finite Coxeter groups and their 0-Hecke algebras, including a two-sided duality between the induced and restricted representations.

Framization of a Temperley-Lieb algebra of type B

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type B. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type B and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type B as a quotient of the Yokonuma-Hecke algebra of type B. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type B to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

Right-angled Coxeter groups, universal graphs, and Eulerian polynomials

We define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W,S), once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G2 constructed from the unlabeled Coxeter graph G of (W,S). When G=Pn is a path with n vertices, we conjecture that G2 is n-universal; this property is equivalent to the existence of an action of R and H(R) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (Pn)2 is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group Un, using results on the Kazhdan–Lusztig basis of H(Un).

K-automorphisms of a weak-crossed product F-algebra over a Galois extension K/F

We compute the K-automorphism group of AK=K⊗FAf of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent part and the reductive part Hˆ. Automorphisms in Hˆ are computed via their restriction to semi-simple part of AK. There is a strong relationship between Hˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of Hˆ, namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.

Integral u-deformed involution modules

Let (W,S) be a Coxeter system and ⁎ an automorphism of W with order ≤2 and S⁎=S. Lusztig and Vogan ([20], [23]) have introduced a u-deformed version Mu of Kottwitz's involution module over the Iwahori–Hecke algebra Hu(W) with Hecke parameter u2, where u is an indeterminate. Lusztig has proved that Mu is isomorphic to the left Hu(W)-submodule of Hˆu generated by X∅:=∑w⁎=w∈Wu−ℓ(w)Tw, where Hˆu is the vector space consisting of all formal (possibly infinite) sums ∑x∈WcxTx (cx∈Q(u) for each x). He speculated that one can extend this by replacing u with any λ∈C∖{0,1,−1}. In this paper, we give a positive answer to his speculation for any λ∈K∖{0,1,−1} and any W, where K is an arbitrary ground field.