Structure of conjugacy classes in Coxeter groups

For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

Bipolar Coxeter groups

We prove that even Coxeter groups, whose Coxeter diagrams contain no (4, 4, 2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application, we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups. © 2014 University of Illinois.

Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.

Conjugacy classes and straight elements in Coxeter groups

It is possible for a group W that is abstractly isomorphic to a Coxeter group to have more than one conjugacy class of Coxeter generating sets, and if S and R are two non-conjugate Coxeter generating sets then it may or may not be the case that some element s ∈ S is conjugate to an element r ∈ R . In this paper we classify the so-called intrinsic reflections: those elements of W whose conjugacy class intersects non-trivially every Coxeter generating set. In combination with previously known results, this leads us to a classification of Coxeter groups for which all Coxeter generating sets are conjugate.

We discuss the theory of certain partially ordered sets that capture the structure of commutation classes of words in monoids. As a first application, it follows readily that counting words in commutation classes is #P-complete. We then apply the partially ordered sets to Coxeter groups. Some results are a proof that enumerating the reduced words of elements of Coxeter groups is #P-complete, a recursive formula for computing the number of commutation classes of reduced words, as well as stronger bounds on the maximum number of commutation classes than were previously known. This also allows us to improve the known bounds on the number of primitive sorting networks.

A generalization of plactic–coplactic equivalences and Kazhdan–Lusztig cells

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

Minimal length elements in some double cosets of Coxeter groups

In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S).

A Coxeter system (W, S ) is called affine-free if its Coxeter diagram contains no affine subdiagram of rank ≥ 3. Let (W, S ) be a Coxeter system of finite rank (i.e. |S | is finite). The main result is that W is affine-free if and only if W has finitely many conjugacy classes of reflection triangles. This implies that the action of W on its Coxeter cubing (defined by Niblo and Reeves [G. Niblo and L. Reeves. Coxeter groups act on CAT(0) cube complexes. J. Group Theory 6 (2003), 399–413]) is cocompact if and only if (W, S ) is affine-free. This result was conjectured in loc. cit. As a corollary, we obtain that affine-free Coxeter groups are biautomatic.

We combinatorially characterize the number cc2 of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. Moreover, we provide formulae for finite and affine types, besides computing cc2 for all triangle groups and RACGs.

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Bar operators for quasiparabolic conjugacy classes in a Coxeter group