1 Introduction

On models of the braid arrangement and their hidden symmetries

Abstract.

The De Concini-Procesi wonderful models of the braid arrangement of type are equipped with a natural action, but only the minimal model admits an ‘hidden’ symmetry, i.e. an action of that comes from its moduli space interpretation. In this paper we explain why the non minimal models don’t admit this extended action: they are ‘too small’. In particular we construct a supermaximal model which is the smallest model that can be projected onto the maximal model and again admits an extended action. We give an explicit description of a basis for the integer cohomology of this supermaximal model.

Furthermore, we deal with another hidden extended action of the symmetric group: we observe that the symmetric group acts by permutation on the set of -codimensionl strata of the minimal model. Even if this happens at a purely combinatorial level, it gives rise to an interesting permutation action on the elements of a basis of the integer cohomology.

1. Introduction

In this paper we focus on two different ‘hidden’ extended actions of the symmetric group on wonderful models of the (real or complexified) braid arrangement. As it is well known, there are several De Concini-Procesi models associated with the arrangement of type (see [DCP1], [DCP2]); these are smooth varieties, proper over the complement of the arrangement, in which the union of the subspaces is replaced by a divisor with normal crossings. Among these spaces there is a minimal one (i.e. there are birational projections from the other spaces onto it), and a maximal one (i.e. there are birational projections from it onto the other spaces). The natural action on the complement of the arrangement of type extends to all of these models.

The first of the two extended actions which we deal with is well known and comes from the following remark: the minimal projective (real or complex) De Concini-Procesi model of type is isomorphic to the moduli space of -pointed stable curves of genus 0, therefore it carries an ‘hidden’ extended action of that has been studied by several authors (see for instance [getzler], [rains2009], [etihenkamrai]).

Now we observe that the action cannot be extended to the non-minimal models (we show this by an example in Section 5).

Why does this happen? This is the first problem discussed in the present paper. We answer to this question by showing in Section 6 that the maximal model is, in a sense, ‘too small’. This takes two steps (see Theorem 6.1):

  1. we identify in a natural way its strata with a subset of 1-codimensional strata of a ‘supermaximal’ model on which the action is defined. This supermaximal model is obtained by blowing up some strata in the maximal model, but it also belongs to the family of models obtained by blowing up building sets of strata in the minimal model; in fact it is the model obtained by blowing up all the strata of the minimal model. The models in are examples of some well known constructions that, starting from a ‘good’ stratified variety, produce models by blowing up a suitable subset of strata (see [procesimacpherson], [li], [gaimrn0] and also [denham] for further references);

  2. we show that the closure of under the action is the set of all the strata of the supermaximal model. More precisely, this means that the supermaximal model is the minimal model in that admits a birational projection onto the maximal model and is equipped with the action.

The second problem addressed by this paper is the computation of the integer cohomology of the complex supermaximal modes described above. The cohomology module provides ‘geometric’ extended representations of and in Theorem 7.2 we exhibit an explicit basis for it. Actually, the statement of Theorem 7.2 is much more general: given any complex subspace arrangement we consider its minimal De Concini-Procesi model and we describe a basis for the integer cohomology of the variety obtained by blowing up all the strata in this minimal model (this variety generalizes in a way the notion of supermaximal model).

We then compute a generating formula for the Poincaré polynomials of the complex supermaximal models of braid arrangements (see Theorems 8.1, 8.2). As a consequence, we also give a formula for the Euler characteristic series in the real case, where Euler secant numbers appear (see Corollary 8.3).

In the last two sections, Section 9 and Section 10, we show that there is another hidden extended action of the symmetric group on the minimal model of a braid arrangement, that is different from the action described above.

In fact, motivated by a combinatorial remark proven in [gaifficayley], we observe that the symmetric group acts by permutation on the set of -codimensionl strata of the minimal model of type .

This happens at a purely combinatorial level and it does not correspond to a geometric action on the minimal model, nevertheless it gives rise to an interesting permutation action on the elements of a basis of the integer cohomology of the complex minimal model. The splitting of these elements into orbits allows us to write (see Theorem 10.1) a generating formula for the Poincaré polynomials of the complex minimal models that is different from the ones available in the literature (see for instance the recursive formula for the Poincaré series computed, in three different ways, in [Manin], [YuzBasi], [GaiffiBlowups]).

2. Wonderful Models

2.1. The Geometric Definition of Building Sets and Nested Sets

In this section we recall from [DCP1], [DCP2] the definitions of building set and nested set of subspaces. Let be a real or complex finite dimensional vector space endowed with an Euclidean or Hermitian non-degenerate product and let be a central subspace arrangement in . For every , we will denote by its orthogonal. We denote by the closure under the sum of and by the arrangement of subspaces in

Definition 2.1.

The collection of subspaces is called building set associated to if and every element of is the direct sum of the maximal elements of contained in (this is called the -decomposition of ).

Given a subspace arrangement , there are several building sets associated to it. Among these there always are a maximum and a minimum (with respect to inclusion). The maximum is , the minimum is the building set of irreducibles that is defined as follows.

Definition 2.2.

Given a subspace , a decomposition of in is a collection () of non zero subspaces in such that

  1. for every subspace such that , we have and .

Definition 2.3.

A nonzero subspace which does not admit a decomposition is called irreducible and the set of irreducible subspaces is denoted by .

Remark 2.1.

As an example, let us consider a root system in (real or complexified vector space) and its associated root arrangement (i.e. is the hyperplane arrangement provided by the hyperplanes orthogonal to the roots in ). In this case the building set of irreducibles is the set whose elements are the subspaces spanned by the irreducible root subsystems of (see [YuzBasi]).

Definition 2.4.

Let be a building set associated to . A subset is called (-)nested, if given any subset (with ) of pairwise non comparable elements, we have that .

2.2. The Example of the Root System

Let or and let us consider the real or complexified root arrangement of type . We think of it as an essential arrangement, i.e. we consider the hyperplanes defined by the equations in the quotient space .

Let us denote by the building set of irreducibles associated to this arrangement. According to Remark 2.1, it is made by all the subspaces in spanned by the irreducible root subsystems. Therefore there is a bijective correspondence between the elements of and the subsets of of cardinality at least two: if the orthogonal of is the subspace described by the equation then we represent by the set . As a consequence, a -nested set is represented by a set (which we still call ) of subsets of with the property that any of its elements has cardinality and if and belong to than either or one of the two sets is included into the other.

As an example we can consider the -nested set represented by the three sets , , . This means that the elements of the nested set are the three subspaces whose orthogonal subspaces in are described respectively by the equations , and .

We observe that we can represent a -nested set by an oriented forest on leaves in the following way. We consider the set . Then the forest coincides with the Hasse diagram of viewed as a poset by the inclusion relation: the roots of the trees correspond to the maximal elements of , and the orientation goes from the roots to the leaves, that are the vertices .

Let us now focus on the maximal building set associated with the root arrangement of type . It is made by all the subspaces that can be obtained as the span of a set of roots. Using the same notation as before, these subspaces can be put in bijective correspondence with the partitions of such that at least one part has cardinality .

For instance,

corresponds to the subspace whose orthogonal is described by the equations and and .

The -nested sets are given by chains of subspaces in (with respect to inclusion). In terms of partitions, this corresponds to give chains of the above described partitions of (with respect to the refinement relation).

One can find in [GaiffiServenti] a description of the maximal model and in [GaiffiServenti2] a description of all the -invariant building sets associated with the root system .

2.3. The construction of wonderful models and their cohomology

In this section we recall from [DCP1] the construction and the main properties of the De Concini-Procesi models.

The interest in these models was at first motivated by an approach to Drinfeld’s construction of special solutions for Khniznik-Zamolodchikov equation (see [drinfeld]). Moreover, in [DCP1] it was shown, using the cohomology description of these models, that the rational homotopy type of the complement of a complex subspace arrangement depends only on the intersection lattice.

Then real and complex De Concini-Procesi models turned out to play a key role in several fields of mathematical research: subspace and toric arrangements, toric varieties (see for instance [DCP3], [feichtneryuz], [rains]), tropical geometry (see [feichtnersturmfels]), moduli spaces and configuration spaces (see for instance [etihenkamrai], [LTV]), box splines, vector partition functions and index theory (see [DCP4], [cavazzanimoci]), discrete geometry (see [feichtner]).

Let us recall how they are defined. We will focus on the case when or . Let be a subspace arrangement in the real or complex space endowed with a non-degenerate euclidean or Hermitian product and let be the complement in of the arrangement . Let be a building set associated to (we can suppose that it contains ). Then one considers the map

where in the first coordinate we have the inclusion and the map from to is the restriction of the canonical projection .

Definition 2.5.

The (compact) wonderful model is obtained by taking the closure of the image of .

De Concini and Procesi in [DCP1] proved that the complement of in is a divisor with normal crossings whose irreducible components are in bijective correspondence with the elements of and are denoted by ().

If we denote by the projection of onto the first component , then can be characterized as the unique irreducible component such that .

A complete characterization of the boundary is then provided by the observation that, if we consider a collection of subspaces in containing , then

is non empty if and only if is -nested, and in this case is a smooth irreducible subvariety obtained as a normal crossing intersection. Sometimes we will denote by with .

The integer cohomology ring of the models in the complex case was studied in [DCP1], where a presentation by generators and relations was provided. The cohomology is torsion free, and in [YuzBasi] Yuzvinski explicitly described some -bases (see also [GaiffiBlowups]). We briefly recall these results (for a description of the cohomology ring in the real case see [rains]).

Let be a building set containing and let us consider a -nested set . We take a subset and an element with the property that for all and we put . As in [DCP1], we define the non negative integer :

Definition 2.6.

Then we consider the polynomial ring where the variables are indexed by the elements of .

Definition 2.7.

Given , , and as before, we define the following polynomial in :

Let us now call by the ideal in generated by these polynomials, for fixed and varying .

Theorem 2.1 (see [Dcp1, Section 5.2]).

Let and be as before, and let us consider the complex model . The natural map , defined by sending to the cohomology class associated to the divisor (restricted to ), induces an isomorphism between and . In particular, in the case , we obtain

Definition 2.8.

Let and be as before. A function is called -admissible if it is or if , is -nested and, for every , .

Now, given a -admissible function , we can consider in the monomial . We will call “-admissible” such monomials.

Theorem 2.2 (see [YuzBasi, Section 3] and [GaiffiBlowups, Section 2]).

The set of -admissible monomials is a -basis for .

2.4. A More General Construction

The construction of De Concini-Procesi models can be viewed as a special case of other more general constructions that, starting from a ‘good’ stratified variety, produce models by blowing up a suitable subset of strata. Among these there are the models described by MacPherson and Procesi in [procesimacpherson] and by Li in [li]. In Li’s paper one can also find a comparison among several constructions of wonderful compactifications by Fulton-Machperson ([fultonmacpherson]), Ulyanov ([Ulyanov]), Kuperberg-Thurston ([Kuperbergthurston]), Hu ([Hu]). A further interesting survey including tropical compactifications can be found in Denham’s paper [denham].

We recall here some basic facts adopting the language and the notation of Li’s paper.

Definition 2.9.

A simple arrangement of subvarieties (or ‘simple stratification’) of a nonsingular variety is a finite set of nonsingular closed subvarieties properly contained in satisfying the following conditions:
(i) the intersection of and is nonsingular and the tangent bundles satisfy ,
(ii) either is equal to some stratum in or is empty.

Definition 2.10.

Let be an arrangement of subvarieties of . A subset is called a building set of if the minimal elements in intersect transversally and the intersection is .

Then, if one has a simple stratification of a nonsingular variety and a building set , one can construct a wonderful model by considering (by analogy with [DCP1]) the closure of the image of the locally closed embedding

where is the blowup of along .

It turns out that

Theorem 2.3 (see [li, Theorem 1.3]).

If one arranges the elements of in such a way that for every the set is building, then is isomorphic to the variety

where denotes the dominant transform of in .

Remark 2.2.

As remarked by Procesi-MacPherson in [procesimacpherson, Section 2.4] it is always possible to choose a linear ordering on the set such that every initial segment is building. We can do this by ordering in such a way that we always blow up first the strata of smaller dimension.

We show two examples that will be crucial in the following sections.

Example 2.1.

In the case of subspace arrangements, the De Concini-Procesi construction and the above construction produce the same models (the only warning is that in the preceding sections a building set was described in a dual way, so the building set of subvarieties is made by the orthogonals of the subspaces in ).

Example 2.2.

Given a De Concini-Procesi model , we notice that its boundary strata give rise to a simple arrangement of subvarieties, and that the set of all strata is a building set. So it is possibile to obtain a ‘model of the model ’. The boundary strata of these ‘models of models’ are indexed by the nested sets of the building set of all the strata of . More precisely, according to the definition given in [procesimacpherson, Section 4], a nested set in this sense is a collection of -nested sets containing linearly ordered by inclusion (we will come back to this, using a more combinatorial definition, in Section 3).

3. Combinatorial Building Sets

After De Concini and Procesi’s paper [DCP1], nested sets and building sets appeared in the literature, connected with several combinatorial problems. In [feichtnerkozlovincidence] building sets and nested sets were defined in the general context of meet-semilattices, and in [delucchinested] their connection with Dowling lattices was investigated. Other purely combinatorial definitions were used to give rise to the polytopes that were named nestohedra in [postnikoreinewilli].

Here we recall the combinatorial definitions of building sets and nested sets of a power set in the spirit of [postnikoreinewilli], [postnikov] (one can refer to [petric2, Section 2] for a short comparison among various definitions and notations in the literature).

Definition 3.1.

A building set of the power set is a subset of such that:

  • If have nonempty intersection, then .

  • The set belongs to for every .

Definition 3.2.

A (nonempty) subset of a building set is a -nested set (or just nested set if the context is understood) if and only if the following two conditions hold:

  • For any we have that either or or .

  • Given elements () of pairwise not comparable with respect to inclusion, their union is not in .

Definition 3.3.

The nested set complex is the poset of all the nested sets of ordered by inclusion.

We notice that actually is a simplicial complex.

Definition 3.4.

If the set has a minimum , the nested set complex is the poset of all the nested sets of containing , ordered by inclusion.

In particular, let us denote by the poset given by the nested sets in that contain .

We observe that any element in can be obtained by the union of with an element of

where and are all the maximal nested sets associated with the building set (and denotes the power set). Given a simplicial complex which is based on some sets (i.e., it is equal to ), Feichtner and Kozlov’s definition of building set of a meet semilattice (see [feichtnerkozlovincidence, Section 2]) can be expressed in the following way: is a building set of if and only if for every the set is a building set of in the sense of Definition 3.1.

Again, according to Feichtner and Kozlov, given a building set of as before, a -nested set is a subset of such that, for every antichain (with respect to inclusion) , the union belongs to .

These definitions of building set and nested sets can be extended in a natural way to . In particular, the maximal building set of is itself. As we observed in Section 2.3, the strata of are indexed by the elements of and, as we remarked in Section 2.4, the set of all these strata is a building set in the sense1 of [procesimacpherson] [li] and [gaimrn0], therefore we can construct the corresponding (real or complex) variety .

Translating into these combinatorial terms the definition given in [procesimacpherson, Section 4], the strata of the variety are indexed by the nested sets of containing in the following way: a stratum of codimension is indexed by where each belongs to and

4. The geometric extended action on

We recall that there is a well know ‘extended’ action on the De Concini-Procesi (real or complex) model : it comes from the isomorphism with the moduli space and the character of the resulting representation on cohomology has been computed in [rains2009] in the real case, and in [getzler] in the complex case.

In order to describe how acts on the strata of it is sufficient to show the corresponding action on , since these strata are indexed by the elements of .

Let be a basis for the root system of type (we added to a basis of the extra root ). We identify in the standard way with the group which permutes and with the transposition . Therefore , the subgroup generated by , is identified with the subgroup which permutes .

Let be a subspace in different from , let and let us consider the subspace according to the natural action of on . Morover we denote by the subspace generated by all the roots of that are orthogonal to . We notice that if some of the roots contained in have in their support then belongs to . Therefore we define the action of on as follows. Let and . We set and for we define

Let us now write explicitly how the above described action extends to . Let

be an element of , i.e. a nested set in that contains and let . Moreover, let us suppose that, for every subspace , the subspace doesn’t belong to , while the subspaces belong to . Then . As one can quickly check, .

Remark 4.1.

This action can also be lifted to the minimal spherical model of type (see for instance the exposition in [callegarogaiffi3, Section 3]).

5. The Extended Action on Bigger Models: the Example of

From now on the minimal and the maximal models associated with the root system will play a special role in this paper. Hence it is convenient to single out them by a new notation.

Definition 5.1.

We will denote by the minimal model and by the maximal model

It is known that is not possible to extend the action from the strata of the boundary of to the strata of the boundary of the non minimal models (see for instance [GaiffiServenti2, Remark 5.4]).

Now we want to construct a model which is ‘bigger’ than (i.e. it admits a birational projection onto ) and is equipped with an action. We will call supermaximal a model which is minimal among the models that have these properties. Let us construct this by an example in the case .

Example 5.1.

We consider the action of the group on the model : the transposition maps the -dimensional strata and as follows:

and

As one of the steps in the construction of , we blow up along the intersection

Hence, in order to have a model with an extended action, we also need to blow up along the intersection

Actually, because of the symmetry, one has to blow up in all the points , where the two roots span an irreducible root subsystem.

Let us denote by the model obtained as the result of all these blowups: one can immediately check that coincides with and therefore the action on described in Section 4 extends to . In the next two sections we will prove that is a supermaximal model for every .

6. The action on and the minimality property of

As explained in Section 3 the strata of are in correspondence with the elements of . The action on the open part of can be extended to the boundary. In fact one can immediately check that the geometric action on can be extended to : let be and element of and let , then sends to where, for every , according to the action on illustrated in the end of Section 4. From the inclusions it immediately follows that , therefore belongs to .

Now we address the following combinatorial problem: what is the minimal building set in that is closed under the action and ‘contains’ ? We start by expressing in a precise way what we mean with ‘contains’ .

Recall that we write for the poset given by the nested sets in that contain , i.e. the poset that indicizes the strata of .

Proposition 6.1.

There is a graded poset embedding of into .

Proof.

Let be an element in . Then is a nested set of the building set containing . This means that its elements are linearly ordered by inclusion: . Now we can express every as the direct sum of some irreducible subspaces , i.e. elements of (). We notice that, for every , the sets (with and, for every , ) is nested in . The map defined by

if , otherwise

is easily seen to be a poset embedding. ∎

Given a complex of nested sets , we will denote by the subset made by the nested sets of cardinality .

The restriction of to is an embedding of into . Now can be identified with (the identification maps , with a nested set of that strictly contains , to ) and we still call the embedding from to . More explicitly, if is a subspace which is the direct sum of the irreducible subspaces then

Theorem 6.1.

The minimal building subset of which contains the image and is closed under the action is itself.

Proof.

Let us consider a building subset of that contains and is closed under the action. We will prove the claim by showing that .

This can be done by induction on the depth of an element of , which is defined in the following way: let be a -nested set that contains and consider the levelled graph associated to . This graps is an oriented tree: it coincides with the Hasse diagram of the poset induced by the inclusion relation, where the leaves are the minimal subspaces of and the root is and the orientation goes from the root to the leaves. A vertex is in level if the maximal length of a path that connects to a leaf is . We say that has depth if is the highest level of this tree. 2

Now we prove by induction on that every element in with depth belongs to .

When this is immediate: given , then is the nested set of depth 1 whose elements are and the maximal elements of contained in . In this way one can show that all the elements of with depth 1 that contain belong to .

Let us check the case . One first observes that, in view of the definition of the action, every nested set of depth of the form , where , belongs to since it can be obtained as for a suitable choice of and of a the nested set of depth . Now we show that also all the nested sets of depth of the form , with and for every , belong to . In fact we can obtain as a union, for every , of the nested sets that belong to as remarked above. Since all these sets have a nontrivial intersection and is building in the Feichtner-Kozlov sense (see Section 3), this shows that belongs to .

Then let us consider a nested set of depth ( ), where

  • is in level ;

  • is not included in ;

  • all the ’s are in level .

This nested set is in since it can be obtained as a union of the nested sets (with depth ) and , where is any nested subset of with depth . We notice that and are in , and have nonempty intersection, therefore their union belongs to .

Now we can show that in there are all the nested sets of depth : if the set has depth , where and the subspaces are in level , while the ’s are in level , we can obtain as the union of the nested sets (for every ) that have pairwise nonempty intersection and belong to , as we have already shown.

Let us now consider and suppose that every nested set in with depth belongs to . Let be a nested set of depth . Let us denote by the nested set obtained removing from the subspaces in level : it belongs to by the inductive hypothesis. Then we consider the nested set obtained removing from the levels : since has depth 2 it belongs to again by the inductive hypothesis. We observe that and have nonempty intersection, therefore their union belongs to . ∎

7. Supermaximal models and cohomology

7.1. The Model is a Supermaximal Model

We can now answer to the question, raised in Section 5, about how to construct a model that is ‘bigger’ than the maximal model, admits the extended action and is minimal with these properties.

Let us state this in a more formal way. Let us consider the poset that indicizes the strata of the minimal model , and let us denote by the family of the models obtained by blowing up all the building subsets of these strata. We observe that has a natural poset structure given by the relation if and only if (by Li’s definition, this also means that there is a birational projection of onto ).

Let us denote by the set the elements of with depth 1. From Theorem 2.3 and Remark 2.2 it follows that if we blowup in the strata that correspond to the elements of (in a suitable order, i.e. first the strata with smaller dimension) we obtain the model .

Definition 7.1.

The supermaximal model associated with the root arrangement is the minimal model in the poset that admits the action and such that .

We notice that this last property means that the supermaximal model admits a birational projection onto .

As a consequence of Theorem 6.1 we have proven the following result:

Theorem 7.1.

The model is the supermaximal model associated with the root arrangement .

Remark 7.1.

There is a family of -invariant building sets that are intermediate between and (these building sets have been classified in [GaiffiServenti2]). Let be such a building set and let be the corresponding model. We will denote by the minimal model in among the models that admit the -action and such that . Depending on the choice of , it may be . This happens for instance when is the building set that contains and all the triples such that the sum of and is direct and has dimension (this building set is denoted by in [GaiffiServenti2]).

7.2. The Cohomology of a Complex Supermaximal Model

The discussion in the preceding sections points out the interest of the supermaximal models and of the corresponding symmetric group actions.

Let be the building set of irreducible subspaces associated with a subspace arrangement in a complex vector space of dimension . In Section 3 we defined the building set , the building set for the supermaximal model for . By analogy with this notation we write for the building set , that generalizes in a way the idea of supermaximal model.

Let be the model obtained by blowing up all the strata of the minimal model . We recall from Example 2.2 that the strata of are in bijection with the nested sets in , according to the constructions given in [procesimacpherson, li].

Let us denote by the projection from onto .

Theorem 7.2.

A basis of the integer cohomology of the complex model is given by the following monomials:

where

  1. is a chain of -nested sets (possibly empty, i.e. ), with ;

  2. the exponents , for , satisfy the following inequalities: , where we put ;

  3. belongs to (if ) or to (if ) and is the image, via