An Explicit Description of Coxeter Homology Complexes

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.


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 Ê about the nonexistence of odd torsion and provides a basis for H * Y G , 2 .
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 M A Φ /Ê 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 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 lowdimensional cases A 3 , A 4 , B 3 and F 4 ; we count cells and write the resulting homology groups which of course are in accordance with the more general results of 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: 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 For instance, the arrangement in Ê2 given by three distinct lines {l 1 , l 2 , l 3 } is not building, while {l 1 , l 2 } 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 {U 1 , . . ., U k }, k ≥ 2, of pairwise noncomparable elements, we have that

Wonderful Models: constructions over Ê
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 map 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 .De Concini and Procesi proved that the complement 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:

ISRN Geometry
Furthermore, given two subspaces H, C ⊂ V , we will denote by If we now denote by π the projection onto the first component P V , D G is equal to the closure of 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}, then 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 Ê n the n − 1th dimensional unit sphere in Ê n , and, for every subspace A ⊂ Ê n , let S A A ∩ S Ê n .Then we can consider the compact manifold and notice that there is an open embedding This is obtained as a composition of the section s : with the map 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 /Ê .
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 /Ê in the following sense: if we denote by cπ the projection onto the first component S Ê n , then cπ is surjective and it is an isomorphism on the preimage of M A /Ê .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 is 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:
3.9 Let T be a subset of A which includes {0}; then: 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 : Let us recall the definition of F: the model CY A is embedded in Now, given any A ∈ A, we can consider the natural isomorphism between A ⊥ and Ê n /A provided by the projection.

ISRN Geometry
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 Ê n /A in particular this means that on the first factor we are considering the projection S Ê n → P Ê n .

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 : 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 A mΦ and CY A mΦ , 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/Ê 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.

Cellular Complexes for A n
Let us first focus on the essential braid arrangement of dimension n: it consists of the hyper- . These hyperplanes are to the roots of the root system A n .
In this section we will describe the minimal spherical model CY mA n and the minimal real model Y mA n associated to this root system.This example has another well-known geometric interpretation, as Y mA n 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 mA n is a quotient of CY mA n , we first give a description of CY mA n as a cell complex, and then we will present the identification map.

The Model CY mA n
In the model CY mA n , 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 σ 1 , . . ., 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 Ê n 1 /Ê x σ i 1 • • • x σ j } 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 d 1 , . . ., 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.

ISRN Geometry
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 mA n 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 p 1 are included in parentheses p for example, 2, 1 ⊂ 2, 1, 3 , we say that p 1 < p 2 ; b if p 1 and p 2 are disjoint, we say that p 1 < p 2 if and only if the greatest number contained in p 1 is smaller than the greatest number contained in p 2 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 p 1 < • • • < 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 where the sum is taken over all the possible parentheses p that can be added to c.

The Model Y mA n
Our next step is to define an identification between cells of the model CY mA n , in order to get Y mA n 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 where k is the number of elements in parentheses p.More explicitly, 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 mA n as a quotient of the cellular complex for CY mA n .
Remark 5.1.We can associate to a cell c the ordered set of its elements s c σ 1 , . . ., σ n 1 forgetting the parentheses data .Since a cell c in Y mA n corresponds to an equivalence class c of cells in CY mA n , we can choose as a representative for c the cell c ∈ c with the smaller associated set s c , according to the lexicographical order.

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 Ê 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 .

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 Λ 1 < • • • < 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 /Ê with its embedding S M A mΦ ⊂ Ê 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 ν w L,Λ times the orientation induced by c on its boundary.
So the boundary of c is where the sum is taken over all Λ ⊂ Δ such that L ∪ {Λ} is still admissible.

The Minimal Model Y mΦ
Now we define the identification of the cells of the model CY mΦ  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.

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 Λ 1 < • • • < 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.

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 A 3 , A 4 , B 3 , B 4 and the minimal model of F 4 .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 .

Definition 3 . 1 .
The compact model Y G is obtained by taking the closure of the image of i.
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 .
Theorem 3.6 see 12 .If one restricts F to CY A , one obtains a surjective mapF : CY A −→ Y A .3.13Remark 3.7.In particular, when S {0}, this statement reduces to the obvious observation that F restricted to M A /Ê is a 2-sheeted covering of P M A .
x j k and has nontrivial intersection with the closure of the chamber C if and only if it is in the form {x σ i 1 , . .., σ n 1 , we can think of c as the closure in CY mA n of the embedding of C/Ê .An irreducible subspace is given by the equation x j 1 • • •

Table 1 :
Description of the fundamental cells.

Table 2 :
Minimal models: number of cells.
. 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 .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 ∂.

Table 4 :
Maximal models: number of cells.If two cells c and c are antipodal, the above relation means