Permutonestohedra

Permutonestohedra

Abstract

There are several real spherical models associated with a root arrangement, depending on the choice of a building set. The connected components of these models are manifolds with corners which can be glued together to obtain the corresponding real De Concini-Procesi models. In this paper, starting from any root system with finite Coxeter group and any -invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) that lie inside the chambers of the arrangement. The main point of this realization is that the convex hull of these nestohedra is a larger polytope, a permutonestohedron, equipped with an action of or also, depending on the building set, of . The permutonestohedra are natural generalizations of Kapranov’s permutoassociahedra.

1 Introduction

Let be an euclidean vector space of dimension , let be a root system which spans and has finite Coxeter group , and let be a -invariant building set associated with (see Section 2 for a definition of building set).

The main goal of this paper is to provide an explicit linear realization of the permutonestohedron , a polytope whose face poset was introduced in [20]; this polytope is linked with the geometry of the wonderful model of the arrangement and is equipped with an action of or also, depending on the building set, of .

Let us briefly describe the framework from which the permutonestohedra arise.

Given a building set one can construct as in [7] a real and a complex De Concini-Procesi model (resp. and ), or their ‘spherical’ version, a real model with corners (see [19]).

These models play a relevant role in several fields of mathematical research: subspace and toric arrangements, toric varieties (see for instance [9], [17], [31]), moduli spaces of curves and configuration spaces (see for instance [13], [26]), box splines and index theory (see the exposition in [8]), discrete geometry (see [14] for further references).

From the point of view of discrete geometry and combinatorics the spherical model is a particularly interesting object. In fact has as many connected components as the number of chambers and these connected components are non linear realizations of some polytopes that belong to the family of nestohedra. The nestohedra have been defined and studied in [28], [29], [35], and successively in several other papers, due to the interest of nested sets complexes in combinatorics and geometry (see for instance [16] for applications to tropical geometry and [4] for applications to the study of Dahmen-Micchelli modules).

Moreover we remark that a suitable gluing of the connected components of produces the model , which is therefore presented as a CW complex. If is -invariant, (as well as ) inherits an action of which gives rise to geometric representations in cohomology.

The permutonestohedron arises in the middle of this rich algebraic and geometric picture. Its name comes from the following remarks:

  • some of its facets lie inside the chambers of the arrangement; they are nestohedra, and their union is a -invariant linear realization of ; their convex hull is the full permutonestohedron;

  • if and is the minimal building set associated to , the permutonestohedron is a realization of Kapranov’s permutoassociahedron (see [25]).

We remark that a non linear realization in of (and of a convex body whose face poset is the same as the face poset of ) was provided in [20].

As we explain in Section 3.3, the linear realization of that we describe in this paper is inspired by (and generalizes in a way) Stasheff and Shnider’s construction of the associahedron in the Appendix B of [33].

The main point in our choice of the defining hyperplanes is that the every nestohedron in lies inside a chamber of the arrangement; furthermore, each of these hyperplanes is invariant with respect to the action of a parabolic subgroup. These are the reasons why, in the global construction, when we consider the convex hull of all the nestohedra that lie in the chambers, the extra facets of the permutonestohedron appear. These turn out to be combinatorially equivalent to the product of a nestohedron with some smaller permutonestohedra (see Theorem 6.1).

We notice that when , as the building set varies, the nestohedra we obtain inside every orthant are all the nestohedra in the ‘interval simplex-permutohedron’ (see [27]). In particular in the case of the associahedron our construction coincides with the one in [33], while for the other nestohedra it is analogue to the constructions in [12] and [11], [3], that are remarkable generalizations of Stasheff and Shnider’s construction.

In general, once is fixed, the nestohedron which lies inside a chamber depends on . For instance, the minimal building set associated to is the building set made by the irreducible subspaces, i.e., the subspaces spanned by the irreducible root subsystems, while the maximal building set is made by all the subspaces spanned by some of the roots. Now, if is , or and is the minimal building set, the nestohedron is a dimensional Stasheff’s associahedron; if is and is the minimal building set, the nestohedron is a graph associahedron of type . For any -dimensional root system, if is the maximal building set the nestohedron is a dimensional permutohedron (so in this case our permutonestohedron could also be called ‘permutopermutohedron’). When the root system is of type , , , the full poset of invariant building sets has been described in [21].

As mentioned before, we obtain the permutonestohedron by taking the convex hull of the nestohedra that lie in the chambers and whose union realizes . From this point of view the construction of the permutonestohedron is inspired by Reiner and Ziegler construction of Coxeter associahedra in [32], since these are obtained as the convex hull of some Stasheff’s associahedra that lie inside the chambers. Anyway we remark that only when and is the minimal building set the permutonestohedron is combinatorially equivalent to a Coxeter associahedron (i.e., in the case of Kapranov’s permutoassociahedron). For instance in the case the Coxeter associahedron is the convex hull of some dimensional Stasheff’s associahedra, while our minimal permutonestohedron is the convex hull of some dimensional Stasheff’s associahedra; furthermore, non minimal permutonestohedra are convex hulls of different nestohedra.

Figure 1: The maximal permutonestohedron of type : inside each chamber of the arrangement there is a nestohedron (in this case a hexagon, i.e. a two dimensional permutohedron), and the permutonestohedron is the convex hull of all these nestohedra.

We will also focus on the group of isometries of the permutonestohedron : it contains or even, depending on the ‘symmetry’ of , the group , and we will describe some conditions which are sufficient to ensure that it is equal to . We will also point out that a subposet of the minimal permutonestohedron of type is equipped with an ‘extended’ action of .

Finally, here it is a short outline of this paper. In Section 2 we briefly recall the basic properties of building sets, nested sets and spherical models, while Section 3 contains the definition of permutonestohedra and the statement of main theorems, Theorems 3.1 and 3.2, whose proofs can be found respectively in Section 4 and Section 5.

Section 6 is devoted to a presentation of the face poset of the permutonestohedron ; this points out the action of , since the faces of are indicized by pairs of the form (coset of , labelled nested set). Furthermore, Theorem 6.1 shows that every facet that does not lie inside a chamber is combinatorially equivalent to the product of a nestohedron with some smaller permutonestohedra. We also show (Corollary 6.2) that is a simple polytope if and only if is the maximal building set associated to .

The full group of isometries that leave invariant is explored in Section 7: if is invariant with respect to this group contains , and we describe some natural conditions that are sufficient to ensure that it is equal to .

In Section 8 we show how the well know ‘extended’ action on the minimal De Concini-Procesi model of type can be lifted to a subposet of the minimal permutonestohedron of type , providing geometrical realizations of the representations , where is any subgroup of the cyclic group of order .

Finally in Section 9 we show some examples and pictures of permutonestohedra and, as an example of face counting, we compute the -vectors of the minimal and of the maximal permutonestohedron of type .

All the pictures of three-dimensional polytopes (like the one in Figure 1) have been realized using the mathematical software package polymake ([22]).


2 Building sets, nested sets, nestohedra and spherical models

Let us recall from [7], [6] the definitions of nested set and building set of subspaces; more precisely, we specialize these definitions to the case we are interested in, i.e. the case when we deal with a central hyperplane arrangement.

Let be a central line arrangement in an Euclidean space . We denote by the closure under the sum of and by the hyperplane arrangement

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 hyperplane 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. In the case of a root arrangement the building set of irreducibles is the set of subspaces spanned by the irreducible root subsystems of the given root system (see [34]).

Definition 2.2.

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

After De Concini and Procesi’s paper [7], nested sets and building sets appeared in the literature, connected with several combinatorial problems. One can see for instance [15], where building sets and nested sets were defined in the more general context of meet-semilattices, and [10] where the connection with Dowling lattices was investigated. Other purely combinatorial definitions were used to give rise to the polytopes that were named nestohedra in [29] and turned out to play a relevant role in discrete mathematics and tropical geometry. Presentations of nestohedra can be found for instance in [28], [29], [35], [16], [2], [3], [27].

Here we recall, for the convenience of the reader, the combinatorial definitions of building sets and nested sets as they appear in [28], [29]. One can refer to Section 2 of [27] for a short comparison among various definitions and notations in the literature.

Definition 2.3.

A building set is a set of subsets of , such that:

  • If have nonempty intersection, then .

  • The set belongs to for every .

Definition 2.4.

A subset of a building set is a nested set if and only if the following three 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 .

  • contains all the sets of that are maximal with respect to inclusion.

The nested set complex is the poset of all the nested sets of ordered by inclusion. A nestohedron is a polytope whose face poset, ordered by reverse inclusion, is isomorphic to the nested set complex .

Let us now consider a ‘geometric’ building set of subspaces associated with a root arrangement, according to the Definition 2.1, and let us suppose that .

Definition 2.5.

We will denote by the building set made by the subspaces in that are spanned by some subset of the set of simple roots.

Now gives rise to a building set in the sense of the Definition 2.3 in the following way: we associate to a subspace the set of indices of the simple roots contained in it.

Having established this correspondence, the nested sets in the sense of De Concini and Procesi are nested sets in the sense of the Definition 2.4 provided that they contain , the maximal subspace in . These nested sets form a nested set complex denoted by . The nestohedra that will appear in our construction, as facets of permutonestohedra, are the nestohedra associated with the nested set complexes .

Now let be, as above, a central line arrangement in an Euclidean space , and let be the complement in to . In [19] the following compactifications of were defined, in the spirit of De Concini-Procesi construction of wonderful models.

Let us denote by the -th dimensional unit sphere in , and, for every subspace , let . Let be a building set associated to , and let us consider the compact manifold

There is an open embedding which is obtained as a composition of the section

with the map

that is well defined on each factor.

Definition 2.6.

We denote by the closure in of .

It turns out (see [19]) that is a smooth manifold with corners. Its (not connected) boundary components are in correspondence with the elements of the building set , and the intersection of some boundary components is nonempty if and only if these components correspond to a nested set. We notice that has as many connected components as the number of chambers of .

In [20] these connected components were realized inside the chambers, as the complements of a suitable set of tubular neighbouroods of the subspaces in , giving rise to some non linear realizations of the nestohedra .

3 The construction of permutonestohedra

3.1 Selected hyperplanes

The goal of this section is to give the equations of the hyperplanes that will be used in the definition of the permutonestohedra.

Let be as before an euclidean vector space of dimension with scalar product . Let us consider a root system in with finite Coxeter group , and a basis of simple roots for . If is irreducible, we consider in the open fundamental chamber the vector

where is the set of positive roots and the simple weights are defined by

(see [1], [24] as general references on root systems and finite Coxeter groups).

If is not irreducible and splits into the irreducible subsystems we put where, for every , is the semisum of all the positive roots of .

Let us denote by the building set of all the subspaces that can be generated as the span of some of the roots in and by the building set of all the irreducible subspaces in . As we recalled in Section 2, is made by all the subspaces that are spanned by the irreducible root subsystems of .

Let be a building set associated to the root system which contains and is -invariant; when the root system is of type , , , these building sets have been classified in [21].

Given , we will denote by the parabolic subgroup generated by the reflections with . We denote by the orthogonal projection of to ; we have

and we also write where belongs to .

Remark 3.1.

If is an irreducible root subsystem, we notice that is the semisum of all its positive roots. If is not irreducible, and splits into the irreducible subsystems , then where is the semisum of all the positive roots of .

Let be the set of indices made by the such that and let be the set of indices, in the complement of , such that iff for some . Then

with all . Therefore

with all .

Remark 3.2.

We put . Despite appearances, this notation will not be confusing.

Let us consider two subspaces in of dimension respectively and write and as non negative linear combinations of the simple roots. We denote by the maximum coefficient of and by the minimum coefficient of and put .

We then define as the maximum among all the with as above.

Definition 3.1.

A list of positive real numbers will be suitable if for every .

We are now in position to define a set of selected hyperplanes that depend on the choice of a suitable list and are indicized by the elements of . These hyperplanes, together with their images via the action, will be the defining hyperplanes of the permutonestohedron .

The motivation for this definition of suitable list will be pointed out in Section 3.3.

We start from:

Definition 3.2.

Let us denote by the hyperplane

and by the closed half-space that contains the origin and whose boundary is .

For every we call the intersection ; all the vectors lie on the hyperplane and their convex hull, as it is well known, is a ()- simplex.

Definition 3.3.

For every , we define the hyperplane as

and we denote by the closed half-space that contains the origin and whose boundary is .

Now we have to define the hyperplanes indicized by the elements of :

Definition 3.4.

Let be a subspace in , i.e. a subspace which does not belong to and is generated by some of the simple roots, and let where is its -decomposition. Then we put

where . We denote by the closed half-space that contains the origin and whose boundary is .

3.2 Definition of the permutonestohedra

This section starts with the definition of the permutonestohedron as the intersection of closed half-spaces. We then state two theorems that show that the permutonestohedron is a convex hull of nestohedra and explicitly determine its vertices.

Definition 3.5.

The permutonestohedron is the polytope given by the intersection of the half-spaces and of the half-spaces , for all , and .

Remark 3.3.

We are in fact defining infinite permutonestohedra, depending on the choice of a suitable list , so we should write instead than . We use the shorter notation since in this paper the dependence on the suitable list will be relevant only when we will deal with the automorphism group in Section 7, where we will take care of avoiding ambiguities.

Now, given a subset of containing and of cardinality , with the property that the vectors are linearly independent, we denote by the vector defined by

As we will observe in Proposition 4.2, every maximal nested set of has the above mentioned property.

The following theorems will be proven in Section 4 and Section 5.

Theorem 3.1.

The intersection of with all the half-spaces , for , is a realization of the nestohedron which lies in the open fundamental chamber. Its vertices are the vectors where ranges among the maximal nested sets in .

Theorem 3.2.

The permutonestohedron coincides with the convex hull of all the nestohedra () that lie inside the open chambers. Its vertices are , where and ranges among the maximal nested sets in .

3.3 Motivations for the choice of suitable lists

In the proofs of Theorems 3.1 and 3.2 the properties of suitable lists will play a crucial role. This is why we devote this section to suitable lists: we will prove a key lemma and we will show how these lists generalize Stasheff and Shnider’s choice of parameters in their construction of the associahedron in [33].

The following lemma introduces an important inequality that is satisfied by suitable lists and is tied to the combinatorics of root systems.

Lemma 3.1.

Let be a suitable list of positive numbers. Let be a subspace in that can be expressed as a sum of some subspaces in (), and let this sum be non-redundant, i.e. if we remove anyone of the subspaces the sum of the others is strictly included in . Then we have

Proof.

Let . We notice that by definition of the numbers we have:

where means that the difference can be expressed as a non negative linear combination of the simple roots. Let be the maximum among the dimensions of the ’s. Then because of the non-redundancy of the ’s: anyone of the ’s contains at least a simple root which is not contained in the others. Now by definition, and then recursively we obtain

Since and (when ), this implies:

The final inequality

is now straightforward. ∎

Depending on and on the building set , there may be lists of numbers that are not suitable but can be used to construct permutonestohedra, since they ensure that the claim of the above lemma is true.

More generally, we could construct permutonestohedra using the following suitable functions: first, for every set of subspaces as in the statement of the Lemma 3.1, let us choose numbers such that

Definition 3.6.

A function is suitable if, for every set of subspaces in as above and for every , it satisfies

This is the essential property we need in our construction of permutonestohedra. As it is shown by the Lemma 3.1, given a suitable list of numbers one obtains a suitable function by putting for every .

We chose to use suitable lists to make our construction more concrete and to obtain more symmetry: since the associated suitable function depends only on the dimension of the subspaces, if is -invariant the automorphism group of the permutonestohedron includes , as we will show in Section 7. Anyway, the definition of the hyperplanes , , and the proofs of Theorem 3.1 and Theorem 3.2 in the next sections could be repeated almost verbatim using a suitable function instead of a suitable list.

A further interesting aspect of suitable lists and suitable functions is illustrated by the example of the root system , that corresponds to the boolean arrangement.

In this case our definition of suitable list consists in the condition for every . As for suitable functions, in their definition we can choose all the numbers equal to 1. Now let us number from 1 to the positive roots of . Then let us denote by the -invariant building set made by the subspaces that are spanned by a set of positive roots whose associated numbers are an interval in . For this we have and the corresponding nestohedron that we obtain in the fundamental chamber is a Stasheff’s associahedron. In fact in this case one immediately checks that our suitable functions are the same as the suitable functions used by Stasheff and Shnider in their construction of the associahedron in Appendix B of [33].

In Section 9 of [12] Došen and Petrić describe a generalization of Stasheff and Shnider’s construction to all the nestohedra that are in the ‘interval simplex-permutohedron’. These nestohedra are exactly all the nestohedra obtained as varies among the -invariant building sets associated to and our suitable functions for this root system are compatible with the ones described in [12].

4 The nestohedron

This section is devoted to the proof of Theorem 3.1. We will give a self-contained proof that the hyperplanes for define a realization of the nestohedron ; as we have remarked in Section 3.3, our construction is a generalization of Stasheff and Shnider’s construction of the associahedron in Appendix B of [33]. We notice that other constructions of nestohedra can be found for instance in [2], [3], [12], [27], [28], [29], [35].

Our choice of the hyperplanes ensures that the resulting nestohedron lies inside a chamber of the arrangement. Furthermore for every the hyperplane is fixed by the action of . Thanks to these properties when we pass to the global construction, and consider the convex hull of all the nestohedra which lie in the chambers, the extra facets of the permutonestohedron appear.

Proposition 4.1.

Let us consider a subset of containing and of cardinality such that the vectors are linearly independent: if is not nested the vector does not belong to .

Proof.

If doesn’t belong to the open fundamental chamber, it does not belong to . In fact let us write a vector in terms of the basis , ; since belongs to for every simple root , we have , which implies for every . 2

Let us then consider the case when lies in the open fundamental chamber.

Let be a subspace in that can be expressed as a sum of some (more than 1) subspaces in . Such exists since is not nested. Now let with , , be a non-redundant sum (see the statement of Lemma 3.1).

First we show that . Since is in the open fundamental chamber, it can be written as with all the coefficients . Then if we have and . According to the definition of the numbers we have that

Then we deduce that , which is a contradiction since by Lemma 3.1.

So we can assume . We will show that .

We notice that

Let us then check if belongs to . As we observed before we have

Now Lemma 3.1 implies , which proves that does not belong to (so it does not belong to ).

Proposition 4.2.

If is a nested set of the vectors are linearly independent. If is a maximal nested set, the vectors are a basis of and the vectors lie inside the fundamental chamber.

Proof.

It is sufficient to prove the linear independence for maximal nested sets, since every nested set can be completed to a maximal nested set.

Let then be a maximal nested set (therefore it contains ). As we have already observed, the vectors are linearly independent if and only if the vectors are linearly independent. If these vectors are not linearly independent, since for every the vector belongs to , we can find a minimal such that the vectors are not linearly independent. Therefore we have, by nestedness of and by minimality of , a relation of the form

This is a contradiction, since contains a simple root which is not contained in any 3: when is written in terms of the simple roots , its coefficient is >0, while when we write () in terms of the simple roots the coefficient is equal to 0.

This proves the linear independence, and therefore is well defined.

Let us now prove that lies in the fundamental chamber. Let us consider the graph associated to . This graph is a tree and coincides with the Hasse diagram of the poset induced by the inclusion relation. Therefore it can be considered as an oriented rooted tree: the root is and the orientation goes from the root to the leaves, that are the minimal subspaces of . We observe that we can partition the set of vertices of the tree into levels: level 0 is made by the leaves, and in general, level is made by the vertices such that the maximal length of an oriented path that connects to a leaf is .

Let . Since is maximal, it contains at least a subspace of dimension 1. In particular, all the minimal subspaces, i.e. the leaves of the graph, have dimension 1. Let be the subspaces of dimension 1 in . From the relation we deduce that for every . Now if is a subspace which in the graph is in level 1, then it contains some of the leaves, say . By the maximality of we deduce that and we can write where is the only simple root which belongs to but does not belong to the leaves of the graph. Then

with and for every . Therefore

From Lemma 3.1 we know that

Then must be and from this we deduce, given that and is a positive multiple of , that . In a similar way, by induction on the level, we prove that all the coefficients are .

Proposition 4.3.

Let us consider a maximal nested set of . For every the vector belongs to the open part of , therefore is a vertex of .

Proof.

We know by definition that belongs to the hyperplanes for every , therefore to prove the claim it is sufficient to show that for every the vector belongs to the open part of .

The set is not nested since is a maximal nested set and doesn’t belong to .

Let be the minimal element in which strictly contains (it could be ). We observe that there is one (and only one, by the maximality of ) simple root which belongs to but doesn’t belong to any such that . Then must belong to : if , we have . We notice that strictly includes the subspaces such that because of the minimality of . Now, since 4 this implies that which contradicts the nestedness of .

Since we can find a subset of such that

and the sum is non-redundant. Then we have

which implies

Now since belongs to the open fundamental chamber (Proposition 4.2) we have

We observe that belongs to the open part of if and only ; this inequality is verified since, by Lemma 3.1 we have:

The proof of Theorem 3.1 is an immediate consequence of the Propositions 4.1, 4.2, 4.3: the faces of dimension are in bijection with the nested sets of cardinality containing .

5 From nestohedra to the permutonestohedron

This section is devoted to the proof of Theorem 3.2. We will split the proof in two steps, given by following propositions:

Proposition 5.1.

For every , for every and for every maximal nested set of , we have

and the equality holds if and only if . This means that belongs to , and it lies in if and only if and ; more precisely, is the affine span of such vectors. Furthermore we have

for every different from the identity.

Proof.

Let us consider (in the case the proof is similar). Let be the set of simple roots that belong to . Then with all the (Remark 3.1).

Since lies inside the open fundamental chamber, we can write with all the . As it is well known (see for instance [1], [24]),

where the are positive roots and : as a notation, when we put and the sum is empty. Then we have:

The scalar product

is easily seen to be since .

If then all the roots belong to , therefore

Otherwise, at least one of the positive roots, say , does not belong to ,5 and we have:

It remains to prove that is the affine span of the vectors with and . The vectors with are all the vertices of a ()-dimensional face of the nestohedron . The affine span of this face is the subspace whose elements satisfy the relations and . Then coincides with the hyperplane defined by the relation .

Now the vectors with and lie on and, since for every simple root which belongs to , we have that is a non zero scalar multiple of , the affine span of these vectors coincides with . ∎

Lemma 5.1.

Let be a subspace which does not belong to and is generated by some of the simple roots and let be its -decomposition. Then the subspaces are pairwise orthogonal. The vector of Definition 3.4 can be obtained as

Furthermore, if is the set of indices given by the such that , we have