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ISRN Geometry
Volume 2011 (2011), Article ID 387936, 13 pages
http://dx.doi.org/10.5402/2011/387936
Research Article

An Explicit Description of Coxeter Homology Complexes

1Scuola Normale Superiore di Pisa, 56126 Pisa, Italy
2Dipartimento di Matematica, Università di Pisa, 56127 Pisa, Italy

Received 12 May 2011; Accepted 21 June 2011

Academic Editors: A. Cattaneo and A. Morozov

Copyright © 2011 Filippo Callegaro and Giovanni Gaiffi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Rains (2010) computes the integral homology of real De Concini-Procesi models of subspace arrangements, using some homology complexes whose main ingredients are nested sets and building sets of subspaces. We think that it is useful to provide various different descriptions of these complexes, since they encode relevant information about the homotopy type of the models and there still are interesting open questions about -bases of the homology modulo its torsion (see the work by Rains (2010)). In this paper we focus on the case of the Coxeter arrangements: we give an explicit and elementary description, in terms of the combinatorics of the Coxeter groups, of the cells and of the boundary maps of these complexes.

1. Introduction

Let 𝒜 be a central subspace arrangement in an euclidean vector space 𝑉 of dimension 𝑛, and let us denote its complement by (𝒜). In [1] De Concini and Procesi construct models for (𝒜), associated with distinct sets of initial combinatorial data (“building sets,” see Section 2) which are subspace arrangements with complement (𝒜).

Let 𝒢 be a building set as above: in [2] Rains computes the integral homology of the real De Concini-Procesi model 𝑌𝒢, using some homology complexes whose main combinatorial ingredients are the nested sets (again see Section 2) of subspaces in 𝒢. In particular, Rains proves the conjecture (formulated in [3] for the particular case of the moduli space 𝑀0,𝑛()) about the nonexistence of odd torsion and provides a basis for 𝐻(𝑌𝒢,𝔽2).

We think that it is useful to provide various different descriptions of these complexes, since they encode relevant information about the homotopy type of the model and there still are interesting open questions about -bases of the homology modulo its torsion (see Section  6 of [2]).

In this paper we focus on the case of the Coxeter arrangements: we give an explicit and elementary description, in terms of the combinatorics of the Coxeter groups, of the cells and of the boundary maps of the complexes associated to the minimal and to the maximal real De Concini-Procesi model (among the building sets associated to a given subspace arrangement there always are a minimal one and a maximal one with respect to inclusion).

Let then 𝑊 be a Coxeter group, and let Φ be its root system, which spans the euclidean space 𝑉. We denote by 𝒜(Φ) the arrangement made by the hyperplanes orthogonal to Φ.

As a specialization of a construction in [4], we consider some models for the 𝐶 manifold (𝒜(Φ))/+ which are 𝐶 compact manifolds with corners. Again they are associated with building sets and their connected components are (diffeomorphic to) polytopes.

Let 𝒢(Φ) be a building set associated with 𝒜(Φ), and let 𝐶𝑌𝒢(Φ) be the related model with corners; according to a “gluing” map described in [4], we obtain the De Concini-Procesi model 𝑌𝒢(Φ) as a quotient of 𝐶𝑌𝒢(Φ) (for different point of views which lead to the same construction, see [58]).

The natural 𝐶𝑊 structure of 𝐶𝑌𝒢(Φ) arising from the stratification of the polytopes in the boundary induces, gluing in a suitable way the faces of the polytopes, a 𝐶𝑊 structure on 𝑌𝒢(Φ).

We describe in detail the resulting homology complex. In particular, in Section 5 we deal, as a first step, with the minimal De Concini-Procesi model associated to the braid arrangement of dimension 𝑛 (that is to say, the 𝐴𝑛 case), which is isomorphic to the real moduli space of genus 0, stable, (𝑛+2)-pointed curves. In Section 6 we study the minimal and maximal models for the general Coxeter groups.

Our description points out (which in fact is the aim of the present paper) how these homology complexes connect the combinatorics of nested sets with the partitions of the Coxeter diagrams and the action of the parabolic subgroups of 𝑊.

In the last section, as a concrete example, we focus on some complexes in low-dimensional cases (𝐴3, 𝐴4, 𝐵3 and 𝐹4); we count cells and write the resulting homology groups which of course are in accordance with the more general results of [2].

2. Building Sets and Nested Sets

Let us rewrite in our euclidean case some definitions from [1]. We start by a (central) subspace arrangement 𝒜 in the euclidean space 𝑉. It is convenient to deal also with its “dual” object: let us denote by 𝒜 the arrangement made by the subspaces orthogonal to the subspaces of 𝒜:𝒜=𝐵.𝐵𝒜(2.1) Then we denote by 𝒞𝒜 the dual of the lattice (𝒜) of intersections of the subspaces in 𝒜; in other words, 𝒞𝒜 is the closure, under the sum, of 𝒜.

In the sequel building, arrangements will play a crucial role.

Definition 2.1. The subspace arrangement 𝒜 in 𝑉 is called “building set” or “building arrangement” if every element 𝐶 of 𝒞𝒜 is the direct sum 𝐶=𝐺1𝐺2𝐺𝑘 of the set of the maximal elements 𝐺1,𝐺2,,𝐺𝑘 of 𝒜 contained in 𝐶.

For instance, the arrangement in 2 given by three distinct lines {𝑙1,𝑙2,𝑙3} is not building, while {𝑙1,𝑙2} is building.

Let be any subspace arrangement in 𝑉; the family of building arrangements that have the same intersection lattice as (in particular, all these arrangements have the same complement in 𝑉) is not empty. Furthermore, in this family there is a minimum and a maximum element with respect to inclusion (which may eventually coincide in trivial cases, see [1]). The elements of the minimum building arrangement are the “irreducible subspaces” of (), while the maximum building set is () itself.

We can now recall the notion of “nested set” (see [1]) which generalizes the one introduced by Fulton and MacPherson in their paper [9] on models of configuration spaces.

Definition 2.2. Let 𝒦 be a building arrangement of subspaces in 𝑉. A subset 𝒮𝒦 is called “nested relative to 𝒦,” or 𝒦-nested, if, given any of its subset {𝑈1,,𝑈𝑘}, 𝑘2, of pairwise noncomparable elements, we have that 𝑘𝑖=1𝑈𝑖𝒦 (or equivalently, 𝑘𝑖=1𝑈𝑖𝒦).

3. Wonderful Models: constructions over

A model for the complement (𝒢) of a subspace arrangement 𝒢 in a real or complex vector space 𝑉, from the point of view of algebraic geometry, is a smooth irreducible variety 𝑌𝒢 equipped with a proper map 𝜋𝑌𝒢𝑉 such that (i)𝜋 is an isomorphism on the preimage of (𝒢);(ii)the complement of this preimage is a divisor with normal crossings.

In their paper [1], De Concini and Procesi constructed models of this type, provided that the set of subspaces 𝒢 is building, and computed their cohomology in the complex case.

In [1] arrangements of linear subspaces in the projective space 𝐏(𝑉) have also been studied: the associated compact models are constructed in the following way.

Let 𝒢 be a building set (we can suppose that it contains {0}), and let 𝐏((𝒢)) be the complement in 𝐏(𝑉) of the projective subspaces 𝐏(𝐴) (𝐴𝒢). Then one considers the map𝑖𝐏((𝒢))𝐏(𝑉)×𝐷𝒢{0}𝐏𝑉𝐷,(3.1) where in the first coordinate we have the inclusion and the map from (𝒢) to 𝐏(𝑉/𝐷) is the restriction of the canonical projection (𝑉𝐷)𝐏(𝑉/𝐷).

Definition 3.1. The compact model 𝑌𝒢 is obtained by taking the closure of the image of 𝑖.

De Concini and Procesi proved that the complement 𝒟 of 𝐏((𝒜)) in 𝑌𝒢 is the union of smooth irreducible divisors 𝒟𝐺 indexed by the elements 𝐺𝒢{0}.

To be more precise, let us introduce the following notation.

Definition 3.2. Given a subspace 𝐶𝑉, we define the following two (possibly empty) subspace arrangements: (1)𝒜𝐶={𝐻𝒜𝐶𝐻}, (2)𝒜𝐶={𝐵𝐶𝐵𝒜𝒜𝐶}.
Furthermore, given two subspaces 𝐻,𝐶𝑉, we will denote by 𝒜𝐶𝐻 the subspace arrangement 𝒜𝐶𝐻={𝐵𝐶𝐵𝒜𝐻(𝒜𝐶𝒜𝐻)}.

If we now denote by 𝜋 the projection onto the first component 𝐏(V), 𝒟𝐺 is equal to the closure of𝜋1𝐏(𝐺)𝐿𝒜𝐺𝐏(𝐿).(3.2) It can also be characterized as the unique irreducible component such that 𝜋(𝒟𝐺)=𝐏(𝐺). A complete characterization of the boundary is provided by the observation that if we consider a collection 𝒯 of subspaces in 𝒢{0}, then𝒟𝒯𝐴𝒯𝒟𝐴(3.3) is nonempty if and only if 𝒯 is nested, and in this case 𝒟𝒯 is a smooth irreducible subvariety.

From the point of view of differentiable geometry, the compact differentiable models of configuration spaces which appear in Kontsevich’s paper [10] on deformation quantization of the Poisson manifolds raised the interest in the construction of differentiable models with corners of real subspace arrangements.

Kontsevich’s compactifications have been shown in [4] (see also [11]) to be particular cases of the following more general construction.

Let us denote by 𝑆(𝑛) the 𝑛1th dimensional unit sphere in 𝑛, and, for every subspace 𝐴𝑛, let 𝑆(𝐴)=𝐴𝑆(𝑛). Then we can consider the compact manifold𝐾=𝑆(𝑛)×𝐴𝒜{0}𝑆𝐴(3.4) and notice that there is an open embedding𝜙(𝒜)+𝐾.(3.5) This is obtained as a composition of the section 𝑠(𝒜)/+(𝒜) provided by𝑠([𝑝]𝑝)=||𝑝||𝑆(𝑛)(𝒜)(3.6) with the map(𝒜)𝑆(𝑛)×𝐴𝒜{0}𝑆𝐴,(3.7) where on each factor we have a well-defined projection.

Definition 3.3. We define 𝐶𝑌𝒜 as the closure in 𝐾 of 𝜙((𝒜)/+).

In [4] it has been proven that when 𝒜 is a building set, 𝐶𝑌𝒜 is a smooth manifold with corners.

It is a differentiable model for (𝒜)/+ in the following sense: if we denote by 𝑐𝜋 the projection onto the first component 𝑆(𝑛), then 𝑐𝜋 is surjective and it is an isomorphism on the preimage of (𝒜)/+. Furthermore, 𝑐𝜋 establishes a bijective correspondence between the (closures of) codimension 1 open strata in the boundary of 𝐶𝑌𝒜 and the elements of 𝒜{0}.

More precisely, if 𝐴𝒜{0}, its associated boundary component is𝐶𝒟𝐴=𝑐𝜋1𝑆(𝐴)𝐵𝒜𝐴.𝑆(𝐵)(3.8)

We notice that the combinatorial structure of the boundary mimicks the one of complex De Concini-Procesi models (see [4]).

Theorem 3.4. 𝐶𝒟𝐴 is a manifold with corners of the following type: 𝐶𝒟𝐴𝐶𝑌𝒜𝐴𝐴×𝐶𝑌𝒜𝐴.(3.9) Let 𝒯 be a subset of 𝒜 which includes {0}; then: 𝐶𝒟𝒯=𝐵𝒯{0}𝐶𝒟𝐵(3.10) is nonempty if and only if 𝒯 is nested in 𝒜.

The relations between the algebraic-geometric and the differentiable construction of models have been studied in [12] by describing the combinatorial properties of a surjective map 𝐹𝐶𝑌𝒜𝑌𝒜.

Let us recall the definition of 𝐹: the model 𝐶𝑌𝒜 is embedded in𝐾=𝑆(𝑛)×𝐴𝒜{0}𝑆𝐴(3.11) while 𝑌𝒜 is embedded inside𝐾=𝐏(𝑛)×𝐷𝒜{0}𝐏𝑛𝐷.(3.12) Now, given any 𝐴𝒜, we can consider the natural isomorphism between 𝐴 and 𝑛/𝐴 provided by the projection.

Remark 3.5. As a consequence of this identification, there is a map 𝐹 from 𝐾 to 𝐾 whose restriction to each factor 𝑆(𝐴) is the 21 projection 𝑆(𝐴)𝐏(𝑛/𝐴) (in particular this means that on the first factor we are considering the projection 𝑆(𝑛)𝐏(𝑛)).

Theorem 3.6 (see [12]). If one restricts 𝐹 to 𝐶𝑌𝒜, one obtains a surjective map 𝐹𝐶𝑌𝒜𝑌𝒜.(3.13) Let 𝒮 be a 𝒜-nested set which contains 0. Then 𝐹 restricted to the internal points of 𝐶𝒟𝒮 is a 2|𝒮|-sheeted covering of the open part of the boundary component 𝒟𝒮 in 𝑌𝒜.

Remark 3.7. In particular, when 𝒮={0}, this statement reduces to the obvious observation that 𝐹 restricted to (𝒜)/+ is a 2-sheeted covering of 𝐏((𝒜)).

4. The Coxeter Arrangements

Let us specialize the results described in the preceding sections to the case of the Coxeter arrangements.

Let 𝑊 be a Coxeter group, and let Φ be its root system, which spans the euclidean space 𝑉.

The arrangement 𝒜(Φ) provided by the hyperplanes orthogonal to the roots is not building in general. In this paper we will restrict our attention to the minimal and maximal building arrangements associated to it: 𝒜𝑚Φ and 𝒜𝑀Φ.

The arrangement 𝒜𝑚Φ is made by the “irreducible” subspaces, that is to say, its elements are the subspaces which are orthogonal to the irreducible root subsystems of Φ (see [13, 14]):𝒜𝑚Φ=𝐽,𝐽Φand𝐽irreducible(4.1) where 𝐽 is the linear span of 𝐽.

The maximal building arrangement 𝒜𝑀Φ is equal to the full lattice of intersections of the hyperplanes orthogonal to the roots. Then, with a slight abuse of notation, we will denote by 𝑌𝑚Φ, 𝐶𝑌𝑚Φ, 𝑌𝑀Φ, and 𝐶𝑌𝑀Φ (instead of by 𝑌𝒜𝑚Φ and 𝐶𝑌𝒜𝑚Φ, etc.) the associated models.

We notice that there is a bijective correspondence between the connected components of 𝐶𝑌𝑚Φ, 𝐶𝑌𝑀Φ (this is true in general for any building set 𝒢(Φ) associated to Φ, not just for the minimal and maximal building sets) and the Weyl chambers. In fact, if 𝐶 is a Weyl chamber, then the closure 𝐶 of the embedding of 𝐶/+ into 𝐶𝑌𝑚Φ (resp., 𝐶𝑌𝑀Φ) is a connected component of 𝐶𝑌𝑚Φ (resp., 𝐶𝑌𝑀Φ).

We also notice that, in general for any building set 𝒢(Φ) associated to the arrangement 𝒜(Φ), the map 𝐹 of Theorem 3.6 is injective when restricted to 𝐶 and 𝐹(𝐶) (and therefore 𝐶) is diffeomorphic to a convex polytope (see [5, 6, 12, 13, 15]). For instance, in the 𝐴𝑛 case, the polytope associated to the minimal building arrangement is a Stasheff’s associahedron (see [16]) while the one associated to the maximum building is a permutohedron. In general for any Φ and any building set 𝒢(Φ), this polytope is a nestohedron (see [1719] and also [20]).

As an immediate consequence, we have the following algebraic-topological corollary of Theorem 3.6, which for simplicity of notation we state for minimal models but which holds for any model.

Corollary 4.1. Let 𝑊 be a Coxeter group with root system Φ, and let 𝑌𝑚Φ and 𝐶𝑌𝑚Φ be as before its associated minimal models. Let us equip 𝐶𝑌𝑚Φ with the 𝐶𝑊 structure provided by the connected components of the open boundary strata; then 𝑌𝑚Φ, with the structure given by the images via 𝐹 of these components, is a 𝐶𝑊 complex and 𝐹 is a map of 𝐶𝑊 complexes.

5. Cellular Complexes for 𝐴𝑛

Let us first focus on the essential braid arrangement of dimension 𝑛: it consists of the hyperplanes {𝑥𝑖=𝑥𝑗} (1𝑖<𝑗𝑛+1) in 𝑉=𝑛+1/111. These hyperplanes are orthogonal to the roots of the root system 𝐴𝑛.

In this section we will describe the minimal spherical model 𝐶𝑌𝑚𝐴𝑛 and the minimal real model 𝑌𝑚𝐴𝑛 associated to this root system. This example has another well-known geometric interpretation, as 𝑌𝑚𝐴𝑛 can be viewed as the real moduli space of genus 0, stable, (𝑛+2)-pointed curves (see [7, 8, 12, 21]). In Section 6 we will see that this construction can be generalized to any Coxeter arrangement. Since the model 𝑌𝑚𝐴𝑛 is a quotient of 𝐶𝑌𝑚𝐴𝑛, we first give a description of 𝐶𝑌𝑚𝐴𝑛 as a cell complex, and then we will present the identification map.

5.1. The Model 𝐶𝑌𝑚𝐴𝑛

In the model 𝐶𝑌𝑚𝐴𝑛, the maximal cells are in correspondence with the elements of the Coxeter group of type 𝐴𝑛, and we denote them by means of the permutation representation on the set {1,,𝑛+1}. So we write 𝑐=(𝜎1,,𝜎𝑛+1) for the (𝑛1)-cell corresponding to the element 𝜎, where 𝜎𝑖=𝜎(𝑖). If we denote by 𝐶 the open chamber in 𝑉=𝑛+1/111 containing the (class of the) vector (𝜎1,,𝜎𝑛+1), we can think of 𝑐 as the closure in 𝐶𝑌𝑚𝐴𝑛 of the embedding of 𝐶/+.

An irreducible subspace is given by the equation 𝑥𝑗1==𝑥𝑗𝑘 and has nontrivial intersection with the closure of the chamber 𝐶 if and only if it is in the form {𝑥𝜎𝑖=𝑥𝜎𝑖+1==𝑥𝜎𝑗} with 𝑖<𝑗 and 𝑗𝑖<𝑛. It follows that we can denote the corresponding cell in the boundary of 𝑐 including into (a couple of) parentheses the numbers 𝜎𝑖,,𝜎𝑗. Finally, given some cells 𝑑1,,𝑑𝑘 in the boundary of 𝑐, their intersection is nonempty if and only if the corresponding subspaces form a nested set. This means that the corresponding parentheses are pairwise disjoint or ordered by inclusion.

For example in the spherical model 𝐶𝑌𝑚𝐴3, (2,1,3,4) is a maximal cell and it has dimension 2. The 1-cells in its boundary are ((2,1),3,4),((2,1,3),4),(2,(1,3),4),(2,(1,3,4)),(2,1,(3,4)).(5.1) The 0-cells are (((2,1),3),4),((2,(1,3)),4),(2,((1,3),4)),(2,(1,(3,4))),((2,1),(3,4)).(5.2)

Now we need to fix an orientation on cells. We can do this on the maximal cells by endowing the sphere 𝑆𝑛1 with the positive orientation and (denoting by the complement of the arrangement) requiring the projection 𝑆()𝐶𝑌𝑚𝐴𝑛 to be orientation preserving. For the lower-dimension cell we need to fix an ordering in the set of parentheses. Given a cell 𝑐, we can order its parentheses in the following way: (a)if parentheses 𝑝1 are included in parentheses 𝑝2 (for example, (2,1)(2,1,3)), we say that 𝑝1<𝑝2; (b)if 𝑝1 and 𝑝2 are disjoint, we say that 𝑝1<𝑝2 if and only if the greatest number contained in 𝑝1 is smaller than the greatest number contained in 𝑝2 (for example, (2,3)<(1,4)).

Now we notice that, for any parentheses 𝑝 that we can add to 𝑐, the corresponding cell is in the boundary of 𝑐. Let 𝑐(𝑝) be the cell obtained from 𝑐 adding the parentheses 𝑝 and suppose that 𝑝1<<𝑝𝑘 are the parenthesis of 𝑐. If 𝑝𝑖<𝑝<𝑝𝑖+1, we define the number 𝜈(c,𝑝)=𝑖 as the position (eventually 0) of the last parentheses before 𝑝 in the ordering of the parentheses of 𝑐. We define the orientation on the cell 𝑐(𝑝) as (1)𝜈(𝑐,𝑝) times the natural orientation induced by 𝑐 on its boundary. So the boundary of the cell 𝑐 is given by 𝜕𝑐=𝑝(1)𝜈(𝑐,𝑝)𝑐(𝑝),(5.3) where the sum is taken over all the possible parentheses 𝑝 that can be added to 𝑐.

5.2. The Model 𝑌𝑚𝐴𝑛

Our next step is to define an identification between cells of the model 𝐶𝑌𝑚𝐴𝑛, in order to get 𝑌𝑚𝐴𝑛 as a quotient complex.

Let 𝑐 be a cell, and let 𝑝 be (a couple of) parentheses of 𝑐. In view of Remark 3.5 it suffices to describe the identifying relation between 𝑐 and the cell 𝑐 obtained from 𝑐 by inverting the order of the numbers contained in the parentheses 𝑝 (and so by inverting the order of the numbers of all parentheses contained in 𝑝). We say that 𝑐(1)𝑘+1𝑐,(5.4) where 𝑘 is the number of elements in parentheses 𝑝. More explicitly, 𝑎𝑖1,,𝑎𝑖𝑘(1)𝑘+1𝑎𝑖𝑘,,𝑎𝑖1.(5.5)

Since the ordering relation between parentheses depends only on the elements in the parentheses, it follows immediately that the identification relation is compatible with the boundary map. These relations, according to Corollary 4.1, describe the cellular complex for the model 𝑌𝑚𝐴𝑛 as a quotient of the cellular complex for 𝐶𝑌𝑚𝐴𝑛.

Remark 5.1. We can associate to a cell 𝑐 the ordered set of its elements 𝑠(𝑐)=(𝜎1,,𝜎𝑛+1) (forgetting the parentheses data). Since a cell ̂𝑐 in 𝑌𝑚𝐴𝑛 corresponds to an equivalence class [𝑐] of cells in 𝐶𝑌𝑚𝐴𝑛, we can choose as a representative for [𝑐] the cell 𝑐[𝑐] with the smaller associated set 𝑠(𝑐), according to the lexicographical order.

6. Cellular Complexes for a Coxeter Arrangement

Let (𝑊,Φ) be a Coxeter system. Let ΔΦ be the set of simple roots. We suppose we realize 𝑊 as a reflection group in the real vector space 𝑉=𝑛 spanned by the roots in Φ and consider the corresponding minimal and maximal building arrangements 𝒜𝑚Φ and 𝒜𝑀Φ. We give in the next two subsections a description of the cell complexes for the minimal models 𝐶𝑌𝑚Φ and 𝑌𝑚Φ. Again we first give a description of the model 𝐶𝑌𝑚Φ, and then we obtain 𝑌𝑚Φ as a quotient. In the last subsection we discuss the changes needed to study the case of the maximal models 𝐶𝑌𝑀Φ and 𝑌𝑀Φ.

6.1. The Minimal Model 𝐶𝑌𝑚Φ

The maximal cells of 𝐶𝑌𝑚Φ are in correspondence with the open chambers 𝐶 of the space (𝒜𝑚Φ) (which coincides with the complement of the union of the hyperplanes orthogonal to the roots in Φ). We now choose a set of simple roots Δ and therefore a fundamental 𝐶𝑒 whose walls are in correspondence with Δ. Then we can fix a point 𝑥 in the fundamental chamber and associate to the element 𝑤𝑊 the chamber 𝐶𝑤 containing the point 𝑤(𝑥). So maximal cells for 𝐶𝑌𝑚Φ are in correspondence with the elements of the group 𝑊.

In the minimal building set every irreducible subspace is the invariant set of a parabolic subgroup. Given a subset ΛΔ such that the corresponding graph ΓΛ is a connected subgraph of the Dynkin diagram ΓΔ, we call 𝐼Λ the invariant subspace of the parabolic subgroup 𝑊Λ generated by Λ. Since a generic parabolic subgroup is conjugated to a parabolic subgroup of type 𝑊Λ, we can write a generic invariant subspace in the form 𝐼(𝑤,Λ)=𝑤𝐼Λ for an element 𝑤𝑊 and for a subset ΛΔ such that the graph ΓΛ is connected. Notice that the couple (𝑤,Λ) is not unique.

We will denote a cell in the boundary of the maximal fundamental cell by a couple (𝑒,), where 𝑒 is the identity in 𝑊 and is an admissible set of subsets of Δ, that is:(a)every set Λ is a proper subset of Δ such that ΓΛ is connected;(b)for any two sets Λ,Λ, either one is included in the other or the two subsets are disjoint and the corresponding subgroups 𝑊Λ and 𝑊Λ commute.

Notice that the admissible sets correspond to the fundamental nested sets described in [1].

In analogy with the previous section, we can think of the set Λ as a couple of “parentheses” in the graph Γ (a “tubing,” see [5]).

We will denote by (𝑤,) the cell in 𝐶𝑌Φ which is equal to 𝑤((𝑒,)).

Now we want to give an orientation to the cells 𝐶𝑌Φ; we start by fixing an ordering on the set of roots Φ. Then we consider a cell (𝑤,): we want to fix an ordering on the elements of which depends on 𝑤. Given two sets Λ,Λ, we say that Λ<Λ if one of the following cases occurs:(a)ΛΛ;(b)max(𝑤Λ)<max(𝑤Λ).

Now let Λ, and let Λ1<<Λ𝑘 be the elements of written according to the above described ordering. Suppose that Λ𝑖<Λ<Λ𝑖+1. We define the integer 𝜈𝑤(,Λ)=𝑖.

We are now ready to give an orientation to the cells in 𝐶𝑌𝑚Φ. For the maximal cells (𝑤,), we do this identifying (𝒜𝑚Φ)/+ with its embedding 𝑆((𝒜𝑚Φ))𝑛 and requiring the map 𝑆((𝒜𝑚Φ))𝐶𝑌𝑚Φ to be orientation preserving. If we suppose we have oriented a cell 𝑐=(𝑤,), we can orient a cell 𝑐=(𝑤,{Λ}) with (1)𝜈𝑤(,Λ) times the orientation induced by 𝑐 on its boundary.

So the boundary of 𝑐 is 𝜕𝑐=Λ(1)𝜈𝑤(,Λ)(𝑤,{Λ}),(6.1) where the sum is taken over all ΛΔ such that {Λ} is still admissible.

6.2. The Minimal Model 𝑌𝑚Φ

Now we define the identification of the cells of the model 𝐶𝑌𝑚Φ. Let 𝑤Δ be the longest element of the Coxeter group 𝑊. In general we will write 𝑤Λ for the longest element of the parabolic subgroup 𝑊Λ. Let 𝑐=(𝑤,), and let 𝑐=(𝑤,) be cells in 𝐶𝑌𝑚Φ: the identifying relation is given by 𝑐(1)dim(Λ)𝑐,(6.2) where 𝑤=𝑤𝑤Λ for a set Λ, and the sets {𝐼(𝑤,Λ)Λ} and {𝐼(𝑤𝑤Λ,Λ)Λ} are equal. Notice that these sets are the nested sets associated with the cells 𝑐 and 𝑐, respectively, and that they are equal if and only if the sets {𝐼(𝑒,Λ)Λ} and {𝐼(𝑤Λ,Λ)Λ} are equal. We notice that the above-described identification relations are compatible with the boundary map 𝜕.

Remark 6.1. If two cells 𝑐 and 𝑐 are antipodal, the above relation means 𝑐(1)𝑛𝑐.(6.3)

Remark 6.2. In order to perform explicit computations, it is useful to choose a standard representative 𝑐 for every cell [𝑐]𝑌𝑚Φ. This can be done for instance by fixing a total ordering on the group 𝑊 and, given a class [𝑐], by choosing the representative 𝑐=(𝑤,)[𝑐] such that 𝑤 is the smallest possible.

6.3. The Maximal Models 𝐶𝑌𝑀Φ and 𝑌𝑀Φ

In the maximal case (maximal models appear for instance in [6]; see also [22] for some further references), we will denote a cell in 𝐶𝑌𝑀Φ by a couple (𝑤,), where as before 𝑤 is an element in 𝑊 and is an admissible set of subsets of Δ, but this time the definition of admissible is the following:(a)every set Λ is a proper subset of Δ (notice that ΓΛ does not need to be connected); (b)the sets in are totally ordered by inclusion.

Let Λ, and let Λ1<<Λ𝑘 be the elements of written according to the inclusion ordering. Suppose that Λ𝑖<Λ<Λ𝑖+1. Then we define the integer 𝜈(,Λ)=𝑖 (notice that this time it does not depend on 𝑤).

Now the boundary map can be defined by the same procedure as in the minimal case.

Also the identification of the cells of the model 𝐶𝑌𝑀Φ can be done following the same rules of the preceding subsection.

7. Some Low-Dimensional Examples

As a concrete example of the combinatorics involved in these homology complexes, we describe by Tables 1, 2, 3, 4, and 5 the minimal and maximal models for the root systems of type 𝐴3,𝐴4,𝐵3,𝐵4 and the minimal model of 𝐹4.

tab1
Table 1: Description of the fundamental cells.
tab2
Table 2: Minimal models: number of cells.
tab3
Table 3: Minimal models: integral homology.
tab4
Table 4: Maximal models: number of cells.
tab5
Table 5: Maximal models: integral homology.

We list the total number of cells in the model, the cells in a foundamental chamber, and we compute (we have been assisted by the computer algebra systems Axiom and Aldor) the resulting homology groups. Of course the listed groups are in accordance with the more general results of [2] (for the rational cohomology of the minimal models see also [23]).

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