An Explicit Description of Coxeter Homology Complexes

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 ℤ-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 ℳ(𝒜). In [1] De Concini and Procesi construct models for ℳ(𝒜), associated with distinct sets of initial combinatorial data (“building sets,” see Section 2) which are subspace arrangements with complement ℳ(𝒜).

Let 𝒢 be a building set as above: in [2] Rains computes the integral homology of the real De Concini-Procesi model 𝑌_𝒢, 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 𝑀_0,𝑛(ℝ)) about the nonexistence of odd torsion and provides a basis for 𝐻_∗(𝑌_𝒢,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 ℤ-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 ℳ(𝒜(Φ))/ℝ⁺ 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 𝐶𝑌_𝒢(Φ) be the related model with corners; according to a “gluing” map described in [4], we obtain the De Concini-Procesi model 𝑌_𝒢(Φ) as a quotient of 𝐶𝑌_𝒢(Φ) (for different point of views which lead to the same construction, see [5–8]).

The natural 𝐶𝑊 structure of 𝐶𝑌_𝒢(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 𝑌_𝒢(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 𝐴_𝑛 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^⟂∣𝐵∈𝒜.(2.1) Then we denote by 𝒞_𝒜 the dual of the lattice ℒ(𝒜) of intersections of the subspaces in 𝒜; in other words, 𝒞_𝒜 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 𝒞_𝒜 is the direct sum 𝐶=𝐺₁⊕𝐺₂⊕⋯⊕𝐺_𝑘 of the set of the maximal elements 𝐺₁,𝐺₂,…,𝐺_𝑘 of 𝒜^⟂ contained in 𝐶.

For instance, the arrangement in ℝ² given by three distinct lines {𝑙₁,𝑙₂,𝑙₃} is not building, while {𝑙₁,𝑙₂} is building.

Let ℬ be any subspace arrangement in 𝑉; the family of building arrangements that have the same intersection lattice as ℬ (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 ℒ(ℬ), while the maximum building set is ℒ(ℬ) 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 𝒮⊂𝒦 is called “nested relative to 𝒦,” or 𝒦-nested, if, given any of its subset {𝑈₁,…,𝑈_𝑘}, 𝑘≥2, of pairwise noncomparable elements, we have that ⋂^𝑘𝑖=1𝑈_𝑖∉𝒦 (or equivalently, ∑^𝑘𝑖=1𝑈^⟂𝑖∉𝒦^⟂).

3. Wonderful Models: constructions over ℝ

A model for the complement ℳ(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 𝑌_𝒢 equipped with a proper map 𝜋∶𝑌_𝒢→𝑉 such that (i)𝜋 is an isomorphism on the preimage of ℳ(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(ℳ(𝒢)) be the complement in 𝐏(𝑉) of the projective subspaces 𝐏(𝐴) (𝐴∈𝒢). Then one considers the map𝑖∶𝐏(ℳ(𝒢))⟶𝐏(𝑉)×_{𝐷∈𝒢−{0}}𝐏𝑉𝐷,(3.1) where in the first coordinate we have the inclusion and the map from ℳ(𝒢) to 𝐏(𝑉/𝐷) is the restriction of the canonical projection (𝑉−𝐷)→𝐏(𝑉/𝐷).

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

De Concini and Procesi proved that the complement 𝒟 of 𝐏(ℳ(𝒜)) in 𝑌_𝒢 is the union of smooth irreducible divisors 𝒟_𝐺 indexed by the elements 𝐺∈𝒢−{0}.

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

Definition 3.2. Given a subspace 𝐶⊂𝑉, we define the following two (possibly empty) subspace arrangements: (1)𝒜_𝐶={𝐻∈𝒜𝐶⊂𝐻}, (2)𝒜^𝐶={𝐵∩𝐶∣𝐵∈𝒜−𝒜_𝐶}.
Furthermore, given two subspaces 𝐻,𝐶⊂𝑉, we will denote by 𝒜^𝐶𝐻 the subspace arrangement 𝒜^𝐶𝐻={𝐵∩𝐶∣𝐵∈𝒜_𝐻−(𝒜_𝐶∩𝒜_𝐻)}.

If we now denote by 𝜋 the projection onto the first component 𝐏(V), 𝒟_𝐺 is equal to the closure of𝜋⁻¹⎛⎜⎜⎝𝐏(𝐺)−_{𝐿∈𝒜𝐺}𝐏(𝐿)⎞⎟⎟⎠.(3.2) It can also be characterized as the unique irreducible component such that 𝜋(𝒟_𝐺)=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}, then𝒟_𝒯≡_𝐴∈𝒯𝒟_𝐴(3.3) is nonempty if and only if 𝒯 is nested, and in this case 𝒟_𝒯 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(ℝ^𝑛) the 𝑛−1th dimensional unit sphere in ℝ^𝑛, and, for every subspace 𝐴⊂ℝ^𝑛, let 𝑆(𝐴)=𝐴∩𝑆(ℝ^𝑛). Then we can consider the compact manifold𝐾=𝑆(ℝ^𝑛)×_{𝐴∈𝒜−{0}}𝑆𝐴^⟂(3.4) and notice that there is an open embedding𝜙∶ℳ(𝒜)ℝ⁺⟶𝐾.(3.5) This is obtained as a composition of the section 𝑠∶ℳ(𝒜)/ℝ⁺→ℳ(𝒜) provided by𝑠([𝑝])=𝑝||𝑝||∈𝑆(ℝ^𝑛)∩ℳ(𝒜)(3.6) with the mapℳ(𝒜)⟶𝑆(ℝ^𝑛)×_{𝐴∈𝒜−{0}}𝑆𝐴^⟂,(3.7) where on each factor we have a well-defined projection.

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

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

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

More precisely, if 𝐴∈𝒜−{0}, its associated boundary component is𝐶𝒟_𝐴=𝑐𝜋⁻¹⎛⎜⎜⎝𝑆(𝐴)−_{𝐵∈𝒜𝐴}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. 𝐶𝒟_𝐴 is a manifold with corners of the following type: 𝐶𝒟_𝐴≅𝐶𝑌_{𝒜𝐴⟂𝐴}×𝐶𝑌_𝒜^𝐴.(3.9) Let 𝒯 be a subset of 𝒜 which includes {0}; then: 𝐶𝒟_𝒯=_{𝐵∈𝒯−{0}}𝐶𝒟_𝐵(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 𝐹∶𝐶𝑌_𝒜→𝑌_𝒜.

Let us recall the definition of 𝐹: the model 𝐶𝑌_𝒜 is embedded in𝐾=𝑆(ℝ^𝑛)×_{𝐴∈𝒜−{0}}𝑆𝐴^⟂(3.11) while 𝑌_𝒜 is embedded inside𝐾^=𝐏(ℝ^𝑛)×_{𝐷∈𝒜−{0}}𝐏ℝ^𝑛𝐷.(3.12) Now, given any 𝐴∈𝒜, we can consider the natural isomorphism between 𝐴^⟂ and ℝ^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 2↦1 projection 𝑆(𝐴^⟂)↦𝐏(ℝ^𝑛/𝐴) (in particular this means that on the first factor we are considering the projection 𝑆(ℝ^𝑛)↦𝐏(ℝ^𝑛)).

Theorem 3.6 (see [12]). If one restricts 𝐹^ to 𝐶𝑌_𝒜, one obtains a surjective map 𝐹∶𝐶𝑌_𝒜⟶𝑌_𝒜.(3.13) Let 𝒮 be a 𝒜-nested set which contains 0. Then 𝐹 restricted to the internal points of 𝐶𝒟_𝒮 is a 2^|𝒮|-sheeted covering of the open part of the boundary component 𝒟_𝒮 in 𝑌_𝒜.

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

The arrangement 𝒜_𝑚Φ 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]):𝒜_𝑚Φ=⟨𝐽⟩^⟂∣𝐽⊆Φand𝐽irreducible,(4.1) where ⟨𝐽⟩ is the linear span of 𝐽.

The maximal building arrangement 𝒜_𝑀Φ 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 𝑌_𝑚Φ, 𝐶𝑌_𝑚Φ, 𝑌_𝑀Φ, and 𝐶𝑌_𝑀Φ (instead of by 𝑌_𝒜𝑚Φ and 𝐶𝑌_𝒜𝑚Φ, etc.) the associated models.

We notice that there is a bijective correspondence between the connected components of 𝐶𝑌_𝑚Φ, 𝐶𝑌_𝑀Φ (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/ℝ⁺ into 𝐶𝑌_𝑚Φ (resp., 𝐶𝑌_𝑀Φ) is a connected component of 𝐶𝑌_𝑚Φ (resp., 𝐶𝑌_𝑀Φ).

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 𝐴_𝑛 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 [17–19] 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 𝑌_𝑚Φ and 𝐶𝑌_𝑚Φ be as before its associated minimal models. Let us equip 𝐶𝑌_𝑚Φ with the 𝐶𝑊 structure provided by the connected components of the open boundary strata; then 𝑌_𝑚Φ, 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 𝐴_𝑛

Let us first focus on the essential braid arrangement of dimension 𝑛: it consists of the hyperplanes {𝑥_𝑖=𝑥_𝑗} (1≤𝑖<𝑗≤𝑛+1) in 𝑉=ℝ^𝑛+1/ℝ^11⋅1. These hyperplanes are orthogonal to the roots of the root system 𝐴_𝑛.

In this section we will describe the minimal spherical model 𝐶𝑌_𝑚𝐴𝑛 and the minimal real model 𝑌_𝑚𝐴𝑛 associated to this root system. This example has another well-known geometric interpretation, as 𝑌_𝑚𝐴𝑛 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 𝑌_𝑚𝐴𝑛 is a quotient of 𝐶𝑌_𝑚𝐴𝑛, we first give a description of 𝐶𝑌_𝑚𝐴𝑛 as a cell complex, and then we will present the identification map.

5.1. The Model 𝐶𝑌_𝑚𝐴𝑛

In the model 𝐶𝑌_𝑚𝐴𝑛, the maximal cells are in correspondence with the elements of the Coxeter group of type 𝐴_𝑛, and we denote them by means of the permutation representation on the set {1,…,𝑛+1}. So we write 𝑐=(𝜎₁,…,𝜎_𝑛+1) for the (𝑛−1)-cell corresponding to the element 𝜎, where 𝜎_𝑖=𝜎(𝑖). If we denote by 𝐶 the open chamber in 𝑉=ℝ^𝑛+1/ℝ^11⋅1 containing the (class of the) vector (𝜎₁,…,𝜎_𝑛+1), we can think of 𝑐 as the closure in 𝐶𝑌_𝑚𝐴𝑛 of the embedding of 𝐶/ℝ⁺.

An irreducible subspace is given by the equation 𝑥_𝑗1=⋯=𝑥_𝑗𝑘 and has nontrivial intersection with the closure of the chamber 𝐶 if and only if it is in the form {𝑥_𝜎𝑖=𝑥_𝜎𝑖+1=⋯=𝑥_𝜎𝑗} with 𝑖<𝑗 and 𝑗−𝑖<𝑛. It follows that we can denote the corresponding cell in the boundary of 𝑐 including into (a couple of) parentheses the numbers 𝜎_𝑖,…,𝜎_𝑗. Finally, given some cells 𝑑₁,…,𝑑_𝑘 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 𝐶𝑌_𝑚𝐴3, (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 𝑆^𝑛−1 with the positive orientation and (denoting by ℳ the complement of the arrangement) requiring the projection 𝑆(ℳ)→𝐶𝑌_𝑚𝐴𝑛 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<⋯<𝑝_𝑘 are the parenthesis of 𝑐. If 𝑝_𝑖<𝑝<𝑝_𝑖+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 𝑌_𝑚𝐴𝑛

Our next step is to define an identification between cells of the model 𝐶𝑌_𝑚𝐴𝑛, in order to get 𝑌_𝑚𝐴𝑛 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, ⋯𝑎_𝑖1,…,𝑎_𝑖𝑘⋯≃(−1)^𝑘+1⋯𝑎_𝑖𝑘,…,𝑎_𝑖1⋯.(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 𝑌_𝑚𝐴𝑛 as a quotient of the cellular complex for 𝐶𝑌_𝑚𝐴𝑛.

Remark 5.1. We can associate to a cell 𝑐 the ordered set of its elements 𝑠(𝑐)=(𝜎₁,…,𝜎_𝑛+1) (forgetting the parentheses data). Since a cell 𝑐̂ in 𝑌_𝑚𝐴𝑛 corresponds to an equivalence class [𝑐] of cells in 𝐶𝑌_𝑚𝐴𝑛, 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 Δ⊂Φ be the set of simple roots. We suppose we realize 𝑊 as a reflection group in the real vector space 𝑉=ℝ^𝑛 spanned by the roots in Φ and consider the corresponding minimal and maximal building arrangements 𝒜_𝑚Φ and 𝒜_𝑀Φ. We give in the next two subsections a description of the cell complexes for the minimal models 𝐶𝑌_𝑚Φ and 𝑌_𝑚Φ. Again we first give a description of the model 𝐶𝑌_𝑚Φ, and then we obtain 𝑌_𝑚Φ as a quotient. In the last subsection we discuss the changes needed to study the case of the maximal models 𝐶𝑌_𝑀Φ and 𝑌_𝑀Φ.

6.1. The Minimal Model 𝐶𝑌_𝑚Φ

The maximal cells of 𝐶𝑌_𝑚Φ are in correspondence with the open chambers 𝐶 of the space ℳ(𝒜_𝑚Φ) (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 𝐶_𝑒 whose walls are in correspondence with Δ. Then we can fix a point 𝑥 in the fundamental chamber and associate to the element 𝑤∈𝑊 the chamber 𝐶_𝑤 containing the point 𝑤(𝑥). So maximal cells for 𝐶𝑌_𝑚Φ 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 Λ⊂Δ such that the corresponding graph Γ_Λ is a connected subgraph of the Dynkin diagram Γ_Δ, we call 𝐼_Λ the invariant subspace of the parabolic subgroup 𝑊_Λ generated by Λ. Since a generic parabolic subgroup is conjugated to a parabolic subgroup of type 𝑊_Λ, we can write a generic invariant subspace in the form 𝐼(𝑤,Λ)=𝑤𝐼_Λ for an element 𝑤∈𝑊 and for a subset Λ⊂Δ such that the graph Γ_Λ 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,ℒ), where 𝑒 is the identity in 𝑊 and ℒ is an admissible set of subsets of Δ, that is:(a)every set Λ∈ℒ is a proper subset of Δ such that Γ_Λ is connected;(b)for any two sets Λ,Λ^∈ℒ, either one is included in the other or the two subsets are disjoint and the corresponding subgroups 𝑊_Λ and 𝑊_Λ^′ 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∈ℒ as a couple of “parentheses” in the graph Γ (a “tubing,” see [5]).

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

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

Now let Λ∉ℒ, and let Λ₁<⋯<Λ_𝑘 be the elements of ℒ written according to the above described ordering. Suppose that Λ_𝑖<Λ<Λ_𝑖+1. We define the integer 𝜈_𝑤(ℒ,Λ)=𝑖.

We are now ready to give an orientation to the cells in 𝐶𝑌_𝑚Φ. For the maximal cells (𝑤,∅), we do this identifying ℳ(𝒜_𝑚Φ)/ℝ⁺ with its embedding 𝑆(ℳ(𝒜_𝑚Φ))⊂ℝ^𝑛 and requiring the map 𝑆(ℳ(𝒜_𝑚Φ))→𝐶𝑌_𝑚Φ to be orientation preserving. If we suppose we have oriented a cell 𝑐=(𝑤,ℒ), we can orient a cell 𝑐^=(𝑤,ℒ∪{Λ}) with (−1)^{𝜈𝑤(ℒ,Λ)} times the orientation induced by 𝑐 on its boundary.

So the boundary of 𝑐 is 𝜕𝑐=_Λ(−1)^{𝜈𝑤(ℒ,Λ)}(𝑤,ℒ∪{Λ}),(6.1) where the sum is taken over all Λ⊂Δ such that ℒ∪{Λ} is still admissible.

6.2. The Minimal Model 𝑌_𝑚Φ

Now we define the identification of the cells of the model 𝐶𝑌_𝑚Φ. Let 𝑤_Δ be the longest element of the Coxeter group 𝑊. In general we will write 𝑤_Λ for the longest element of the parabolic subgroup 𝑊_Λ. Let 𝑐=(𝑤,ℒ), and let 𝑐^=(𝑤^,ℒ^) be cells in 𝐶𝑌_𝑚Φ: the identifying relation is given by 𝑐≃(−1)^dim(Λ)𝑐^,(6.2) where 𝑤^=𝑤𝑤_Λ for a set Λ∈ℒ, and the sets {𝐼(𝑤,Λ)∣Λ∈ℒ} and {𝐼(𝑤𝑤_Λ,Λ)∣Λ∈ℒ^} 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∈ℒ} and {𝐼(𝑤_Λ,Λ)∣Λ∈ℒ^} 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]∈𝑌_𝑚Φ. 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,ℒ)∈[𝑐] such that 𝑤 is the smallest possible.

6.3. The Maximal Models 𝐶𝑌_𝑀Φ and 𝑌_𝑀Φ

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

Let Λ∉ℒ, and let Λ₁<⋯<Λ_𝑘 be the elements of ℒ written according to the inclusion ordering. Suppose that Λ_𝑖<Λ<Λ_𝑖+1. Then we define the integer 𝜈(ℒ,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 𝐶𝑌_𝑀Φ 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.

Table 1 

Description of the fundamental cells.

Table 2 

Minimal models: number of cells.

Table 3 

Minimal models: integral homology.

Table 4 

Maximal models: number of cells.

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]).