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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Kazhdan Lusztig Cells in Infinite Coxeter Groups

Belolipetsky MV1,2* and Gunnells PE3

1Department of Mathematical Sciences, Durham University, South Rd, Durham DH1 3LE, UK

2Institute of Mathematics, Koptyuga 4, 630090 Novosibirsk, Russia

3Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA

Corresponding Author:
Belolipetsky MV
Department of Mathematical Sciences
Durham University, South Rd, Durham DH1 3LE
UKInstitute of Mathematics, Koptyuga 4
630090 Novosibirsk, Russia
Tel: +44 191 334 2000
E-mail: mikhail.belolipetsky @durham.ac.uk

Received date: July 21, 2015 Accepted date: August 03, 2015 Published date: August 31, 2015

Citation: Belolipetsky MV, Gunnells PE (2015) Kazhdan Lusztig Cells in Infinite Coxeter Groups. J Generalized Lie Theory Appl S1:002. doi:10.4172/1736-4337.S1-002

Copyright: © 2015 Belolipetsky MV, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Groups defined by presentations of the form â¨s1,...,sn | si2 = 1, (sisj)mij = 1(i,j=1,...,n)â© are called Coxeter groups. The exponents mi,j ∈ N ∪ { ∞ } form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group Sn is isomorphic to the Coxeter group with presentation â¨s1,...,sn | si2 = 1 (i=1,...,n),(sisi+1)3=1(i=1,...,n-1)â©, and is also known as the Weyl group of type An-1.

Keywords

Lusztig cells; Coxeter; Hyperplane

Introduction

The notion of cells was introduced by Kazhdan and Lusztig [1,2] to study representations of Coxeter groups and their Hecke algebras. Later it was realized that cells arise in many different branches of mathematics and have many interesting properties. Some examples and references for the related results can be found in Gunnells et al. [3]. We are interested in the combinatorial structure of the cells in infinite Coxeter groups. Examples of such groups include the affine Weyl groups, as well as Weyl groups of hyperbolic Kac–Moody algebras. In contrast with the affine case, there are very few results on cells in hyperbolic Coxeter groups available so far. We refer to Bedard and Belolipetsky [4,5] for some work in this direction. Based on numerous computational experiments we introduce two conjectures that describe the structure of the cells in infinite Coxeter groups. Our conjectures use combinatorial rigidity for elements in an infinite Coxeter group (§4). We expect that this notion may be of independent interest. We were able to check the conjectures for affine groups of small rank [6] (§5), and to prove them in some special cases [7] (§4). The latter include, in particular, the right-angled Coxeter groups previously studied in Belolipetsky [5]. The proof of the conjectures in their general form remains an open problem.

Visualization of cells

Let W be a Coxeter group with a fixed system of generators S. Consider a real vector space V of dimension |S| with a basis image We define a symmetric bilinear form on V by

image

Now for every s ∈ S we can define a linear map image by image

This map sends αs to −αs and fixes the hyperplane Hs orthogonal to αs with respect to B in V. Therefore, σs is a reflection of the space V. One can show that the map image extends to a faithful linear representation of the group W in GL(V), called the standard geometric realization of the group.

Next let us introduce the Tits cone image of W. Every hyperplane Hs divides V into two halfspaces. Let image denote the closed halfspace on which the element image dual to αs is nonnegative. The intersection of these halfspaces imagefor s ∈ S is a closed simplicial cone in V. The closure of the union of all W -translates of image is again a cone in V. This cone C is called the Tits cone. It is known that C=V if and only if W is finite. For infinite groups W the Tits cone is significantly smaller than the whole space and hence is more convenient for the geometric realization of the group.

Under some additional assumptions we can define the action of W on a section of the Tits cone. In particular, this way we can describe the action of affine or hyperbolic Coxeter groups of rank 3 on a Euclidean or Lobachevsky plane, respectively. For example, consider the affine group W of type image This group is generated by three involutions s1,s2, s3 satisfying the relations image for all i ≠ j . The space V is isomorphic to image and via this the Tits cone C can be identified with the upper halfspace image It is easy to check that the action of W preserves the affine plane image , and that the images of image intersect M in equilateral triangles (Figure 1). This construction implies that the group W of type image can be realized as the discrete subgroup of the affine isometrics of a plane generated by reflections in the sides of an equilateral triangle.

generalized-theory-applications-Slicing-the-Tits-cone

Figure 1: Slicing the Tits cone.

A generalization of this construction leads to triangle groups. Let Δ be a triangle with angles π/p, π/q, π/r, where p, q, r ∈image∪ {∞}. The triangle Δ lives on a sphere, or in an affine or hyperbolic plane, depending on whether 1/p + 1/q + 1/r is > 1, = 1, or < 1. The group Wpqr of isometries of the corresponding space generated by reflections in the sides of Δ is respectively a finite, affine, or hyperbolic Coxeter group. For example W333 is the affine group of type image . In Figure 2 we show the tessellations of the hyperbolic plane corresponding to W237 (the Hurwitz group, Figure 2a and W23∞ (the modular group, Figure 2b). The coloring of the triangles indicates the partition of W into cells, which we will define in the next section.

generalized-theory-applications-W237-and-W23

Figure 2: W237 and W23∞.

Main Definitions

Consider a Coxeter group W with a system of generators S. Any element w ∈W can be written as a product, or word, in the generators image Such an expression is called reduced if we cannot use the relations in W to produce a shorter expression for w.

An element can have different reduced expressions but it is not hard to check that all of them have the same length. Therefore, we can define the length function image, which assigns to an element w ∈W the length of a reduced expression with respect to the generators S [1]. Another important notion which can be defined using the reduced expressions is the partial order ≤ of Chevalley–Bruhat. Let image be a word in the generators. We define a subexpression as to be any (possibly empty) product of the form image where image We say that y ≤ w if an expression for y appears as a subexpression of a reduced expression for w. It can be shown that the relation ≤ is a partial order on the group W [1].

Let H denote the Hecke algebra of W over the ring image of Laurent polynomials in q1/2. This algebra is a free A module with a basis Tw, w ∈ W and with multiplication defined by image if image and image for s ∈ S. Together with the basis image we can define in ï another basis image This new basis, introduced by Kazhdan and Lusztig [2], has a number of important properties and has proven to be very convenient for describing the representations of W and H. The elements Cw can be expressed in terms of Tw by the formulae

image

where the image are the Kazhdan–Lusztig polynomials. The polynomials image are nonzero exactly when y, w ∈W satisfy y ≤ w, equal 1 when y = w, and otherwise have degree deg(Py,w ) at most d(y,w) := (l(w) − l(y) − 1)/2. If deg(Py,w ) = d(y, w), we denote the leading coefficient by μ(y, w) = μ(w, y), and in all other cases (including when y and w are not comparable in the partial order) we put μ(y, w) = μ(w, y) = 0. We indicate that μ(y, w)≠ 0 by y−−w.

Using the polynomials Py,w we can define the partial orders ≤L, ≤R, ≤LR on W. First, for w ∈W we define the left and right descent sets:

image

Next, we say that image if there exists a sequence image in W such that imageandimage for all 0 ≤ i < n . The relation image can be defined using image we put image if image . Finally, y ≤LR w means that there exists a sequenceimage such that for all i<n, we have either image or image It is easy to check that ≤L, ≤R and ≤LR are partial orders on W; we denote the corresponding equivalence relations by ∼L, ∼R and ∼LR. The equivalence classes of ∼L (respectively, ∼R, ∼LR) are called the left cells (resp. right cells, two-sided cells) of W. It follows from the definitions that the left and right cells have very similar properties, thus we will mostly deal with only one of these two types of cells.

The colors in Figure 2 indicate the left cells of W237 and the 2-sided cells of W 23∞. Furthermore, the left cells of W 23∞ can be seen in Figure 2b also: they are the connected unions of triangles of a given color. Note that both of these groups apparently have infinitely many left cells. Now consider the multiplication of the C-basis element in H. We can write

image

Let a(z) be the smallest integer such that image for all x, y ∈ W, where image Denote by Di the set image where δ(z) is the degree of the polynomial Pe,z , l(z) is the length function on W, and a(z) is defined as above. The set D = D0 consists of distinguished involutions of W introduced by Lusztig [8].

Lusztig proved that in affine groups there is a bijection between the set D and the left (or right) cells of W [8]. This deep result has many important corollaries and applications. One of the main goals of our work is to make this correspondence between distinguished involutions and cells explicit, so that one can describe the structure of the cells using the distinguished involutions of the group. Another goal is to find an algorithm that produces distinguished involutions of a given group. In the next section we will formulate two conjectures that answer these questions and present some results to support the conjectures.

Conjectures and Results

We will need some more notations and definitions. Given w ∈ W, we write w = x.y if w = xy and l(w) = l(x) + l(y). Denote by Z(w) the set of all v ∈ W such that w = x.v.y for some x, y ∈ W and v ∈ WI for some I ⊂ S with WI finite. (We recall that for a subset I ⊂ S, WI denotes the standard parabolic subgroup of W generated by s ∈ I.) We call imagemaximal in w if it is not a proper sub word of any other v′∈Z(w) such that image with image and image . Let Z = Z (W) be the union of Z (w) over all w ∈ W, image be the set of distinguished involutions of the finite standard parabolic subgroups of W and image Note that each of the sets Z(W), Df and imageis finite.

We call w = x.v.y rigid at v if (i) v ∈ Df , (ii) v is maximal in w, and (iii) for every reduced expression image with image we have image and image This notion of combinatorial rigidity plays an important role in our considerations. Figure 3 helps to understand its meaning using the Cayley graph of the group W. Maximal distinguished involutions of the finite parabolic subgroups correspond to the “long cycles” in the graph. Combinatorial rigidity means that such a cycle cannot be shifted along the presentation of w in any direction. For example, in the triangle group W333 of type image , the element w = s3s1s2s1s3 is rigid at v = s1s2s1, butimageis not rigid at v (Figure 3).

generalized-theory-applications-expressions-Cayley

Figure 3: (a) Rigid and (b) Non-rigi,expressions in the Cayley graph.

Our conjectures can be formulated as follows:

Conjecture 1: (“distinguished involutions”) Let image with a(v) = a(v1), and letimage and image with s ∈ S. Then if sxv1 is rigid at v1, we have image.

Conjecture 2: (“basic equivalences”) Let w = y.v0 with v0 maximal in w.

(a) Let image satisfies image and image is reduced and has image . Then there exists v01 such that image , is rigid at v1 for every image such that image and image the right descent set image , and image which implies image

(b) Let image with image not maximal in image and image Then we can write image that image is maximal in image, image and image So again image Conjecture 1 can be used to inductively construct distinguished involutions in an infinite Coxeter group W starting from the involutions of its finite standard parabolic subgroups. Conjecture 2, in turn, allows one to obtain equivalences in the group using its distinguished involutions. Let us note that the usual method for obtaining results of this kind is based on computing Kazhdan–Lusztig polynomials. This requires many computations and, moreover, does not give any a priori information about the elements that are distinguished involutions or satisfy the cell equivalences in the group. For infinite Coxeter groups in which an exhaustive search is not possible this latter disadvantage becomes critical. Our Conjecture 2 does not necessarily give all equivalences in the group, but still one can expect that the equivalences provided by the conjecture suffice for describing the cells. More precisely, we have the following theorem.

Theorem 1: If an infinite Coxeter group satisfies Conjectures 1 and 2, and also two conjectures of Lusztig, then [7].

(1) The set D of distinguished involutions consists of the union of v∈Df and the elements of W obtained from them using Conjecture 1.

(2) The relations described in Conjecture 2 determine the partition of W into right cells.

(3) The relations described in Conjecture 2 together with its ∼L-analogue determine the partition of W into two-sided cells.

The conjectures of Lusztig to which we refer in the theorem are socalled “positivity conjecture” and a conjecture about a combinatorial description of the function a(z). The positivity conjecture is now proved for a wide class of infinite Coxeter groups that includes affine Weyl groups. The precise statements of the conjectures and the references for the known results can be found in Belolipetsky et al. [7].

We were able to prove Conjectures 1 and 2 under certain additional assumptions. The results are given in the following two theorems.

Theorem 2: [BG1] Let image with image ,a(v) = a(vs), and image Then if v′= s.v.s is rigid at v1, we have image

Theorem 3: Let image with image, image is the longest element of a standard finite parabolic subgroup of W which is maximal in w and a(w) = a(v0), image with image such that a(u) = a(u0) = l, and image with image has image and image

Assume that

(1) For any image image and image or image if image , we have image , image and image is rigid at v0.

(2) For any image j =1,…,l −1 with u1 = u, we have image , image and image is rigid at u0.

Then image and image

The additional assumption image in Theorem 2 may seem minor, but unfortunately this is not the case. In particular, conditions (1) and (2) in Theorem 3 appear as a consequence of this assumption. The proof of the theorems 2 and 3 in Belolipetsky et al. [7] essentially uses the results from two unpublished letters of Springer and Lusztig [9]. A possible approach to the proof of our conjectures in general requires developing further the ideas of this correspondence.

Although the conditions of Theorems 2 and 3 are not always met for all W, the theorems can still be used to produce interesting results. We will give some examples in the next section, other applications of the theorems are considered in Belolipetsky et al. [7].

Cells in Affine Groups of Rank 3

Affine Weyl groups of rank 3 have type image, image or image. The cells in these groups were first described by Lusztig [10]. In this section we will show how the same results can be relatively easily obtained using conjectures from §4.

Type image : The group W is generated by involutions s1, s2, s3 with relations image We have image , image

Applying Conjecture 1 with v = v1 = s1s3s1, we get image Note that after this the inductive procedure terminates as the elements s1 s2v and s3s2v which would come out on the next step are both non-rigid at v. We can apply the same procedure to the other two involutions from image . As a result we get

image

Therefore, the group W of type image has 10 left (right) cells. Using Conjecture 2 and the geometric realization of the group it is easy to show that the partition of W into cells is the one shown on Figure 4a, where two-sided cells correspond to the regions of the same color and left cells correspond to the connected components of the two-sided cells. This coincides with the result of Lusztig et al., [10]. Note that in order to produce the cells we only need Theorems 2 and 3, and thus our results for this case are unconditional.

generalized-theory-applications-Cells-in-A2

Figure 4: Cells in , and .

Type image The group W is generated by involutions s1, s2, s3 with image

image

Conjecture 1 gives

image

The partition of W into cells, which we get using Conjecture 2, is shown in Figure 4b. One can quite easily obtain this partition by following the cycles which correspond to the distinguished involutions on the tessellation of the plane. The result is again in agreement with Lusztig et al. [10]. Note that in this case one can check that the assumptions of Theorem 2 hold, but not the assumptions of Theorem 3. Thus we can compute the distinguished involutions, but we cannot prove that our relations suffice to generate the cells, and therefore cannot be sure that we actually have all the distinguished involutions without using Lusztig’s computations in Lusztig et al. [10] or referring to our conjectures.

Type image The group W is generated by involutions s1, s2, s3 with

image

image

The application of Conjecture 1 in this case already requires some effort because of a large number of possible variants. In order to generate the list of distinguished involutions we used a computer. Our algorithms and their application to other affine Weyl groups are described in Belolipetsky et al. [6]. As a result of these computations, we obtain

image

Therefore, the group W of type image has 28 left (right) cells. The interested reader can check that Conjecture 2 allows us to obtain the partition of W into cells which is shown on Figure 4c. Note that for this case the conditions of neither Theorem 2 nor 3 are satisfied. For instance, we have

image

Thus our results here rely on unproved instances of the conjectures, but nevertheless the results agree with Lusztig et al. [10].

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