# An Explicit Description of Coxeter Homology Complexes

Authors: Filippo Callegaro; Giovanni Gaiffi
Journal: ISRN Geometry
 (2011)

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## 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 aboutℤ-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.


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## Body

## 1. Introduction

Let𝒜 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).Let𝒢 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 M¯0,n(ℝ)) 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 aboutℤ-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 thenW 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.Let𝒢(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 [5–8]).The naturalCW 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 Section5 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 ofW.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:(2.1)A⊥={B⊥∣B∈A}.
Then we denote by 𝒞𝒜 the dual of the lattice ℒ(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 arrangement𝒜 in V is called “building set” or “building arrangement” if every element C of 𝒞𝒜 is the direct sum C=G1⊕G2⊕⋯⊕Gk of the set of the maximal elements G1,G2,…,Gk of 𝒜⊥ contained in C.For instance, the arrangement inℝ2 given by three distinct lines {l1,l2,l3} is not building, while {l1,l2} is building.Letℬ 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.
Let𝒦 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,…,Uk}, k≥2, of pairwise noncomparable elements, we have that ⋂i=1kUi∉𝒦 (or equivalently, ∑i=1kUi⊥∉𝒦⊥).


## 3. Wonderful Models: constructions overℝ

A model for the complementℳ(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 π:Y𝒢→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.Let𝒢 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 map(3.1)i:P(M(G))⟶P(V)×∏D∈G-{0}P(VD),
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 modelY𝒢 is obtained by taking the closure of the image of i.De Concini and Procesi proved that the complement𝒟 of P(ℳ(𝒜)) in Y𝒢 is the union of smooth irreducible divisors 𝒟G indexed by the elements G∈𝒢-{0}.To be more precise, let us introduce the following notation.Definition 3.2.
Given a subspaceC⊂V, we define the following two (possibly empty) subspace arrangements:  (1)
𝒜C={H∈𝒜C⊂H},(2)
𝒜C={B∩C∣B∈𝒜-𝒜C}.
Furthermore, given two subspacesH,C⊂V, we will denote by 𝒜HC the subspace arrangement 𝒜HC={B∩C∣B∈𝒜H-(𝒜C∩𝒜H)}.If we now denote byπ the projection onto the first component P(V), 𝒟G is equal to the closure of(3.2)π-1(P(G)-⋃L∈AGP(L)).
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}, then(3.3)DT≡⋂A∈TDA
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 byS(ℝn) the n-1th dimensional unit sphere in ℝn, and, for every subspace A⊂ℝn, let S(A)=A∩S(ℝn). Then we can consider the compact manifold(3.4)K=S(Rn)×∏A∈A-{0}S(A⊥)
and notice that there is an open embedding(3.5)ϕ:M(A)R+⟶K.
This is obtained as a composition of the section s:ℳ(𝒜)/ℝ+→ℳ(𝒜) provided by(3.6)s([p])=p|p|∈S(Rn)∩M(A)
with the map(3.7)M(A)⟶S(Rn)×∏A∈A-{0}S(A⊥),
where on each factor we have a well-defined projection.Definition 3.3.
We defineCY𝒜 as the closure in K of ϕ(ℳ(𝒜)/ℝ+).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 forℳ(𝒜)/ℝ+ in the following sense: if we denote by cπ the projection onto the first component S(ℝn), 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 CY𝒜 and the elements of 𝒜-{0}.More precisely, ifA∈𝒜-{0}, its associated boundary component is(3.8)CDA=cπ-1(S(A)-⋃B∈AAS(B))¯.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:
(3.9)CDA≅CYAAA⊥×CYAA.
Let 𝒯 be a subset of 𝒜 which includes {0}; then:
(3.10)CDT=⋂B∈T-{0}CDB
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:CY𝒜→Y𝒜.Let us recall the definition ofF: the model CY𝒜 is embedded in(3.11)K=S(Rn)×∏A∈A-{0}S(A⊥)
while Y𝒜 is embedded inside(3.12)K′=P(Rn)×∏D∈A-{0}P(RnD).
Now, given any A∈𝒜, we can consider the natural isomorphism between A⊥ and ℝn/A provided by the projection.Remark 3.5.
As a consequence of this identification, there is a mapF′ from K to K′ whose restriction to each factor S(A⊥) is the 2↦1 projection S(A⊥)↦P(ℝn/A) (in particular this means that on the first factor we are considering the projection S(ℝn)↦P(ℝn)).Theorem 3.6 (see [12]).
If one restrictsF′ to  CY𝒜, one obtains a surjective map
(3.13)F:CYA⟶YA.
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, when𝒮={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.LetW be a Coxeter group, and let Φ be its root system, which spans the euclidean space V.The arrangement𝒜(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 arrangement𝒜mΦ 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]):(4.1)AmΦ={〈J〉⊥∣J⊆ΦandJirreducible},
where 〈J〉 is the linear span of J.The maximal building arrangement𝒜MΦ 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 ofCYmΦ, 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/ℝ+ into CYmΦ (resp., CYMΦ) is a connected component of CYmΦ (resp., CYMΦ).We also notice that, in general for any building set𝒢(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 [17–19] and also [20]).As an immediate consequence, we have the following algebraic-topological corollary of Theorem3.6, which for simplicity of notation we state for minimal models but which holds for any model.Corollary 4.1.
LetW 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 forAn

Let us first focus on the essential braid arrangement of dimensionn: it consists of the hyperplanes {xi=xj} (1≤i<j≤n+1) in V=ℝn+1/ℝ(11·1). These hyperplanes are orthogonal to the roots of the root system An.In this section we will describe the minimal spherical modelCYmAn 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 ModelCYmAn

In the modelCYmAn, 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 (n-1)-cell corresponding to the element σ, where σi=σ(i). If we denote by C the open chamber in V=ℝn+1/ℝ(11·1) containing the (class of the) vector (σ1,…,σn+1), we can think of c as the closure in CYmAn of the embedding of C/ℝ+.An irreducible subspace is given by the equationxj1=⋯=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 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 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 modelCYmA3, (2,1,3,4) is a maximal cell and it has dimension 2. The 1-cells in its boundary are (5.1)((2,1),3,4),((2,1,3),4),(2,(1,3),4),(2,(1,3,4)),(2,1,(3,4)).
The 0-cells are (5.2)(((2,1),3),4),((2,(1,3)),4),(2,((1,3),4)),(2,(1,(3,4))),((2,1),(3,4)).Now we need to fix an orientation on cells. We can do this on the maximal cells by endowing the sphereSn-1 with the positive orientation and (denoting by ℳ the complement of the arrangement) requiring the projection S(ℳ)→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 parenthesesp1 are included in parentheses p2 (for example, (2,1)⊂(2,1,3)), we say that p1<p2;(b)
ifp1 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 parenthesesp 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 (5.3)∂c=∑p(-1)ν(c,p)c(p),
where the sum is taken over all the possible parentheses p that can be added to c.


### 5.2. The ModelYmAn

Our next step is to define an identification between cells of the modelCYmAn, in order to get YmAn as a quotient complex.Letc 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 (5.4)c≃(-1)k+1c′,
where k is the number of elements in parentheses p. More explicitly, (5.5)(⋯(ai1,…,aik)⋯)≃(-1)k+1(⋯(aik,…,ai1)⋯).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 Corollary4.1, 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 cellc the ordered set of its elements s(c)=(σ1,…,σn+1) (forgetting the parentheses data). Since a cell ĉ 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.


## 5.1. The ModelCYmAn

In the modelCYmAn, 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 (n-1)-cell corresponding to the element σ, where σi=σ(i). If we denote by C the open chamber in V=ℝn+1/ℝ(11·1) containing the (class of the) vector (σ1,…,σn+1), we can think of c as the closure in CYmAn of the embedding of C/ℝ+.An irreducible subspace is given by the equationxj1=⋯=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 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 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 modelCYmA3, (2,1,3,4) is a maximal cell and it has dimension 2. The 1-cells in its boundary are (5.1)((2,1),3,4),((2,1,3),4),(2,(1,3),4),(2,(1,3,4)),(2,1,(3,4)).
The 0-cells are (5.2)(((2,1),3),4),((2,(1,3)),4),(2,((1,3),4)),(2,(1,(3,4))),((2,1),(3,4)).Now we need to fix an orientation on cells. We can do this on the maximal cells by endowing the sphereSn-1 with the positive orientation and (denoting by ℳ the complement of the arrangement) requiring the projection S(ℳ)→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 parenthesesp1 are included in parentheses p2 (for example, (2,1)⊂(2,1,3)), we say that p1<p2;(b)
ifp1 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 parenthesesp 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 (5.3)∂c=∑p(-1)ν(c,p)c(p),
where the sum is taken over all the possible parentheses p that can be added to c.


## 5.2. The ModelYmAn

Our next step is to define an identification between cells of the modelCYmAn, in order to get YmAn as a quotient complex.Letc 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 (5.4)c≃(-1)k+1c′,
where k is the number of elements in parentheses p. More explicitly, (5.5)(⋯(ai1,…,aik)⋯)≃(-1)k+1(⋯(aik,…,ai1)⋯).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 Corollary4.1, 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 cellc the ordered set of its elements s(c)=(σ1,…,σn+1) (forgetting the parentheses data). Since a cell ĉ 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 Δ⊂Φ be the set of simple roots. We suppose we realize W as a reflection group in the real vector space V=ℝn 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 ModelCYmΦ

The maximal cells ofCYmΦ are in correspondence with the open chambers C of the space ℳ(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 w∈W 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 subsetΛ⊂Δ 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 w∈W and for a subset Λ⊂Δ 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,ℒ), where e is the identity in W and ℒ is an admissible set of subsets of Δ, that is:(a)
every setΛ∈ℒ is a proper subset of Δ such that ΓΛ is connected;(b)
for any two setsΛ,Λ′∈ℒ, 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 setΛ∈ℒ as a couple of “parentheses” in the graph Γ (a “tubing,” see [5]).We will denote by(w,ℒ) the cell in CYΦ which is equal to w((e,ℒ)).Now we want to give an orientation to the cellsCYΦ; 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 letΛ∉ℒ, and let Λ1<⋯<Λk be the elements of ℒ written according to the above described ordering. Suppose that Λi<Λ<Λi+1. We define the integer νw(ℒ,Λ)=i.We are now ready to give an orientation to the cells inCYmΦ. For the maximal cells (w,∅), we do this identifying ℳ(𝒜mΦ)/ℝ+ with its embedding S(ℳ(𝒜mΦ))⊂ℝn and requiring the map S(ℳ(𝒜mΦ))→CYmΦ 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 ofc is (6.1)∂c=∑Λ(-1)νw(L,Λ)(w,L∪{Λ}),
where the sum is taken over all Λ⊂Δ such that ℒ∪{Λ} is still admissible.


### 6.2. The Minimal ModelYmΦ

Now we define the identification of the cells of the modelCYmΦ. 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,ℒ), and let c′=(w′,ℒ′) be cells in CYmΦ: the identifying relation is given by (6.2)c≃(-1)dim⁡(Λ¯)c′,
where w′=wwΛ¯ for a set Λ¯∈ℒ, and the sets {I(w,Λ)∣Λ∈ℒ} and {I(wwΛ¯,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(wΛ¯,Λ)∣Λ∈ℒ′} are equal. We notice that the above-described identification relations are compatible with the boundary map ∂.Remark 6.1.
If two cellsc and c′ are antipodal, the above relation means
(6.3)c≃(-1)nc′.Remark 6.2.
In order to perform explicit computations, it is useful to choose a standard representativec 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,ℒ)∈[c] such that w is the smallest possible.


### 6.3. The Maximal ModelsCYMΦ 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,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 setΛ∈ℒ is a proper subset of Δ (notice that ΓΛ does not need to be connected);(b)
the sets inℒ are totally ordered by inclusion.LetΛ∉ℒ, and let Λ1<⋯<Λk be the elements of ℒ written according to the inclusion ordering. Suppose that Λi<Λ<Λi+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 modelCYMΦ can be done following the same rules of the preceding subsection.


## 6.1. The Minimal ModelCYmΦ

The maximal cells ofCYmΦ are in correspondence with the open chambers C of the space ℳ(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 w∈W 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 subsetΛ⊂Δ 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 w∈W and for a subset Λ⊂Δ 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,ℒ), where e is the identity in W and ℒ is an admissible set of subsets of Δ, that is:(a)
every setΛ∈ℒ is a proper subset of Δ such that ΓΛ is connected;(b)
for any two setsΛ,Λ′∈ℒ, 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 setΛ∈ℒ as a couple of “parentheses” in the graph Γ (a “tubing,” see [5]).We will denote by(w,ℒ) the cell in CYΦ which is equal to w((e,ℒ)).Now we want to give an orientation to the cellsCYΦ; 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 letΛ∉ℒ, and let Λ1<⋯<Λk be the elements of ℒ written according to the above described ordering. Suppose that Λi<Λ<Λi+1. We define the integer νw(ℒ,Λ)=i.We are now ready to give an orientation to the cells inCYmΦ. For the maximal cells (w,∅), we do this identifying ℳ(𝒜mΦ)/ℝ+ with its embedding S(ℳ(𝒜mΦ))⊂ℝn and requiring the map S(ℳ(𝒜mΦ))→CYmΦ 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 ofc is (6.1)∂c=∑Λ(-1)νw(L,Λ)(w,L∪{Λ}),
where the sum is taken over all Λ⊂Δ such that ℒ∪{Λ} is still admissible.


## 6.2. The Minimal ModelYmΦ

Now we define the identification of the cells of the modelCYmΦ. 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,ℒ), and let c′=(w′,ℒ′) be cells in CYmΦ: the identifying relation is given by (6.2)c≃(-1)dim⁡(Λ¯)c′,
where w′=wwΛ¯ for a set Λ¯∈ℒ, and the sets {I(w,Λ)∣Λ∈ℒ} and {I(wwΛ¯,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(wΛ¯,Λ)∣Λ∈ℒ′} are equal. We notice that the above-described identification relations are compatible with the boundary map ∂.Remark 6.1.
If two cellsc and c′ are antipodal, the above relation means
(6.3)c≃(-1)nc′.Remark 6.2.
In order to perform explicit computations, it is useful to choose a standard representativec 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,ℒ)∈[c] such that w is the smallest possible.


## 6.3. The Maximal ModelsCYMΦ 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,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 setΛ∈ℒ is a proper subset of Δ (notice that ΓΛ does not need to be connected);(b)
the sets inℒ are totally ordered by inclusion.LetΛ∉ℒ, and let Λ1<⋯<Λk be the elements of ℒ written according to the inclusion ordering. Suppose that Λi<Λ<Λi+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 modelCYMΦ 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 Tables1, 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.
ModelsmA3mB3MA3MB3mA4mB4mF4mD4MA4,MB4MD4MF4# 0-cells56141624# 1-cells56212436# 2-cells1191014# 3-cells00111Table 2
Minimal models: number of cells.
ModelA3A4B3B4D4F4RankC015105303361921008RankC1303156010085763024RankC212270248644802592RankC3060019296576Table 3
Minimal models: integral homology.
ModelA3A4B3B4D4F4H0ℤℤℤℤℤℤH1ℤ4⊕ℤ2ℤ10⊕ℤ26ℤ7⊕ℤ2ℤ22⊕ℤ213ℤ16⊕ℤ213ℤ50⊕ℤ215H20ℤ9⊕ℤ20ℤ21⊕ℤ2ℤ15⊕ℤ2ℤ49⊕ℤ2H3000000Table 4
Maximal models: number of cells.
ModelA3A4B3B4D4RankC01818036576288RankC136540721728864RankC212420241344672RankC3060019296Table 5
Maximal models: integral homology.
ModelA3A4B3B4D4H0ℤℤℤℤℤH1ℤ7⊕ℤ2ℤ25⊕ℤ216ℤ13⊕ℤ2ℤ58⊕ℤ241ℤ34⊕ℤ225H20ℤ24⊕ℤ20ℤ57⊕ℤ2ℤ33⊕ℤ2H300000We 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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*Source: 387936-2011-09-11.xml*