Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $ϕ$ of $W$ over the finite field $F_2$ of two elements. The action of $ϕ(W)$ on $F_2^n$ by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of $F_2^n$ under the action of $ϕ(W)$, and find that the kernel of $ϕ$ is the center $Z(W)$ of $W.$
For a coincidental Coxeter group, i.e. of type $A_{n-1}, BC_n, H_3,$ or $I_2(m)$, we define the corresponding $q$-Kreweras numbers attached to limit symbols in the sense of Shoji. The construction of these numbers resembles the argument of Reiner and Sommers and these two share similar properties, but our version is slightly different from and combinatorially simpler than theirs. We also study the combinatorial properties of our $q$-Kreweras numbers, i.e. positivity, relation with $q$-Narayana numbers, and cyclic sieving phenomenon.
The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it has |W| simple and projective modules. In order to construct a combinatorial model for the simp…
An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is investigated in detail. The resulting representations are completely classified and include the irreducible ones.
As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin diagram automorphisms. Combinatorial properties of twisted AR-quivers are useful to understand not only Dorey's rule but also other notions in the representation theory of the quantum affine algebra $U_q'(C_n^{(1)})$ such as denominator formulas. In addition, unlike twisted adapted classes of type $A_{2n-1}$ in the previous paper, we show twisted AR-quivers…
Asymptotic triangulations can be viewed as limits of triangulations under the action of the mapping class group. In the case of the annulus, such triangulations have been introduced by Baur and Dupont. We construct an alternative method of obtaining these asymptotic triangulations using Coxeter transformations. This provides us with an algebraic and combinatorial framework for studying these limits via the associated quivers.
We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.
Chapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.
For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.
We consider Lusztig's $\mathbf{a}$-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of $\mathbf{a}$-value 2 in terms of Coxeter diagrams.
Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial construction of g, where root vectors correspond to vertices of a certain quiver $Gammahat$, and show that with respect to this basis the structure constants of the Lie bracket are given by paths in $Gammahat$. This construction is then related to the constructions of Ringel and Peng and Xiao.
Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in W under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element c is a Coxeter element if and only if there exists a simple system S of reflections such that c is the product of the generators in S. We moreover deduce multiple furth…
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.
Let $W_a$ be an affine Weyl group and $η:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $η(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold…
We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.
In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahor…
A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $Δ$, there is a bijection from the Coxeter group $W$ of type $Δ$ to the set of semibricks over the preprojective algebra $Π$ of type $Δ$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $Π$. This paper is devoted to giving an explicit description of these bijections in the case $Δ=\mathbb{A}_n$ or $\mathbb{D}_n$. First, for each join-irreducible element $w \i…
In this expository note, I showcase the relevance of Coxeter groups to quiver representations. I discuss (1) real and imaginary roots, (2) reflection functors, and (3) torsion free classes and c-sortable elements. The first two topics are classical, while the third is a more recent development. I show that torsion free classes in rep Q containing finitely many indecomposables correspond bijectively to c-sortable elements in the corresponding Weyl group. This was first established in Dynkin type by Ingalls and Thomas; it was shown in general by Amiot, Iyama, Reiten, and Todorov. The proof in th…
We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group $W$ have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in $W$.
In this paper we show that for a simply-laced root system a choice of $C$ 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 $Ihat$ of $I x Z_{2h}$, where $h$ is the Coxeter number. The set $Ihat$ 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 $Ihat$, and the root lattice and the inner product admit an …
Hecke algebras are beautiful q-extensions of Coxeter groups. In this paper, we prove several results on their characters, with an emphasis on characters induced from trivial and sign representations of parabolic subalgebras. While most of the results in type A are known, our proofs are of a combinatorial nature, and generalize to (partial) results in types B and C. We also present complete descriptions of such characters for type I.
We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial Dynkin diagram automorphism. As applications of the study, we prove that folded AR-quivers encode crucial information on the representation theory of quantum affine algebra $U_q'(B^{(1)}_{n+1})$ such as Dorey's rule and denominator formulas.
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 use a generalization of the Littelmann path model to show the existence of the crystals, and study an associated 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.
Interchanging character and co-character groups of a torus $T$ over a field $k$ introduces a contravariant functor $T \rightarrow \widehat{T}$. Interpreting $ρ:T\rightarrow {\mathbb C}^\times$, half the sum of positive roots for $T$ a maximal torus in a simply connected semi-simple group $G$ (over ${\mathbb C}$) using this duality, we get a co-character $\widehatρ: {\mathbb C}^\times \rightarrow \widehat{T}$ whose value at $e^{\frac{2 πi}{h}}$ ($h$ the Coxeter number) is the Coxeter conjugacy class of the dual group $\widehat{G}$. This point of view gives a rather transparent proof of a theore…
For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.
Motivated by work of Coxeter (1957), we study a class of algebras associated to Coxeter groups, which we term 'generalized nil-Coxeter algebras'. We construct the first finite-dimensional examples other than usual nil-Coxeter algebras; these form a $2$-parameter type $A$ family that we term $NC_A(n,d)$. We explore the combinatorial properties of these algebras, including the Coxeter word basis, length function, maximal words, and their connection to Khovanov's categorification of the Weyl algebra. Our broader motivation arises from complex reflection groups and the Broue-Malle-Rouquier freenes…
Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra $H$ and prove that $T_{w_\co}$, where $\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \cite{L4}.
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …
We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A\_n$ and a Coxeter element. Our study is based on Lusztig's …
Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$ over $\mathbb{C}$, $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $\mathfrak{h}$ whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of $\mathfrak{g}$.
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.
We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by a simple reflection, a study of the right descent set, and a proof that the affine length is preserved along the tower of affine Coxeter groups of type $\tilde A$, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower.
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 $x$ \emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$ is a negative root. We call a positive root \emph{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 (Math. Ann. \textbf{296} (1993), 179--190) that $\E$ is finite if (and only if) $W$ is a finite-rank Coxeter group. Amongst other things, this finiteness …
Let $(W,S,L)$ be a weighted Coxeter system and $J$ a subset of $S$, Yin [12] introduced the weighted $W$-graph ideal $E_J$ and the weighted Kazhdan-Lusztig polynomials $ \left \{ P_{x,y} \mid x,y\in E_J\right \}$. In this paper, we study the combinatorial formulas for $P_{x,y}$, which will extend the results of Brenti [2] and Deodhar [5].
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras: a novel 2-parameter type $A$ family that we call $NC_A(n,d)$. We explore several combinatorial properties of $NC_A(n,d)$, including its Coxeter word basis, length function, and Hilbert-Poincare series, and show that the corresponding generalized Coxeter group is not a flat deformation of $NC_A(n,d)$. These algebras yield symmetric semigroup module categorie…
For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $\mathbb{C}[G^{e,c^2}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_-c^2B_-$ has a structure of cluster algebra of finite type, where $B$ and $B_-$ are opposite Borel subgroups. In this article, we consider the case $G$ is of type ${\rm B}_r$, ${\rm C}_r$ or ${\rm D}_r$ and describe all the cluster variables in $\mathbb{C}[G^{e,c^2}]$ as monomial realizations of certain Demazure crystals.
We prove some injectivity results: that a Coxeter monoid $\mathbb{Z}$-algebra (or $0$-Hecke algebra) injects in the incidence $\mathbb{Z}$-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid $\mathbb{Z}[q,q^{-1}]$-algebra (and then in the incidence $\mathbb{Z}[q,q^{-1}]$-algebra of the corresponding Bruhat poset); that a right-angled Artin group injects in the group of invertible elements of the Hecke algebra of the corresponding Coxeter group (and then in the group of invertible elements of a Co…
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to a cluster algebra $\mathcal{A}(\textbf{i})_{\mathbb C}$ [arXiv:math/0305434, arXiv:1602.00498]. In the case $u=e$, $v=c^2$ ($c$ is a Coxeter element), the algebra ${\mathbb C}[G^{e,c^2}]$ has only finitely many cluster variables. In this article, for $G={\rm SL}_{r+1}(\mathbb{C})$, we obtain explic…
In this paper, we study affine Deligne--Lusztig varieties $X_w(b)$ when the finite part of the element $w$ in the Iwahori--Weyl group is a partial $σ$-Coxeter element. We show that such $w$ is a cordial element and $X_w(b) \neq \emptyset$ if and only if $b$ satisfies a certain Hodge--Newton indecomposability condition. The main result of this paper is that for such $w$ and $b$, $X_w(b)$ has a simple geometric structure: the $σ$-centralizer of $b$ acts transitively on the set of irreducible components of $X_w(b)$; and each irreducible component is an iterated fibration over a classical Deligne-…
