Poincaré series for maximal De Concini-Procesi models of root arrangements

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1 Poincaré series for maimal De Concini-Procesi models of root arrangements Giovanni Gaiffi and Matteo Serventi arxiv:0.5533v [math.at] 7 Feb 0 Introduction January 3, 03 In this paper we focus on maimal comple De Concini-Procesi models associated to root arrangements of types A, B, C, D and we compute inductive formulas for their Poincaré series. In [], [3], De Concini and Procesi constructed wonderful models for the complement of a subspace arrangement in a vector space. In general, given a subspace arrangement, there are several De Concini-Procesi models associated to it, depending on distinct sets of initial combinatorial data ( building sets, see Section.). The interest in these varieties was at first motivated by an approach to Drinfeld construction of special solutions for Khniznik-Zamolodchikov equation (see [6]). Then real and comple De Concini-Procesi models turned out to play a central role in several fields of mathematical research: subspace (and toric) arrangements, toric varieties, moduli spaces of curves, configuration spaces, bo splines, inde theory and discrete geometry (see for instance [4], [5], [7], [8], [9], [], [4], [5] and [9]). Among the building sets associated to a given subspace arrangement there are always a minimal one and a maimal one with respect to inclusion: as a consequence there are always a minimal and a maimal De Concini-Procesi model. Several eamples of minimal models (associated to the minimal building set of irreducible subspaces) have been studied in detail. More recently, the relevance of real and comple maimal models was pointed out (maimal models appear for instance in [3], [7], and in the contet of toric varieties, see [] for further references). The case of root arrangements is particularly interesting. Let us consider for instance the arrangements of type A n. Our purpose is to compute the series φ A (q,t) = t+ n P CAn (q) tn n! Q[q][[t]] Here, for every n, P CAn (q) = (rk ( H i( ))) Y CAn,Z q i is the Poincaré polynomial of the maimal comple De Concini-Procesi model Y CAn and the variable q has degree two (in odd degree, the integer cohomology of De Concini-Procesi models is 0 - see [3]).

2 By carefully counting the elements of a basis for the integer cohomology which was first described by Yuzvinski (see [8] and also [0]), we find an inductively defined series in infinite variables g 0,g,g,...g n,... with the following property: when we replace g 0 with t and, for every i,r, gi r with qr q q tr, we obtain the series φ A (q,t) (Theorem 3.). Some eplicit computations (using the Computer Algebra system Aiom) show that our method is effective (see Section 3.). The same technique can be etended to the case B n (see Theorem 3.3), which, from the point of view of subspaces and models, is equal to C n, and also to the D n case (see Section 3.4). In Section 4 we show that the series in infinite variables computed in the preceding sections encode more general results. For instance, they allow us to obtain the Poincaré series of the families of De Concini-Procesi models whose building sets are the maimal building sets C An,C Bn,C Dn tensored by C h : it is sufficient to perform different substitutions of the variables g 0,g,...g n,... We observe that in the A n case, the complements of these tensored arrangements are classical generalizations of the pure braid space (see [] and [6]). Basic Concepts. Some combinatorics of subspace arrangements Let V be a finite dimensional vector space over an infinite field K and denote by V its dual. Let now G be a finite set of subspaces of V and denote by C G its closure under the sum. Definition.. Given a subspace U C G, a decomposition of U in C G is a collection {U,,U k } (k > ) of non zero subspaces in C G such that. U = U U k. for every subspace A U, A C G, we have A U,,A U k C G and A = (A U ) (A U k ). Definition.. A subspace F C G which does not admit a decomposition is called irreducible and the set of irreducible subspaces is denoted by F G. One can prove (see [3]) that every subspace U C G has a unique decomposition into irreducible subspaces. Definition.3. A collection G of subspaces of V is called building if every element C C G is the direct sum G G k of the set of maimal elements G,,G k of G contained in C. Remark.. (see [3]) The set of irreducible subspaces of a given family of subspaces of V is building. A set of subspaces of V which is closed under the sum is building. Given a family G of subspaces of V there are different sets B of subspaces of V such that C B = C G ; if we order by inclusion the collection of such sets, it turns out that the minimal element is F G and the maimal one is C G.

3 Definition.4. (see [3]) Let G be a building set of subspaces of V. A subset S G is called G-nested if and only if for every subset {A,,A k } (k ) of pairwise non comparable elements of S the subspace A = A + +A k does not belong to G. Remark.. If C is a building family of subspaces closed under the sum, then the subspaces of a C-nested set must be totally ordered (with respect to inclusion).. Wonderful models Let now V be a finite dimensional comple vector space and denote by V its dual. Let us consider a finite subspace arrangement G in V and, for every A G, let us denote by A its annihilator in V. For every A G we have a rational map which is regular on V A. We then consider the embedding π A : V V/A P ( V/A ) φ G : A G V A G P ( V/A ) given by inclusion on the first component and the maps π A on the other components. The De Concini-Procesi model Y G associated to G is the closure of φ G (A G ) in V A G P( V/A ). These wonderful models are particularly interesting when the arrangement G is building: they turn out to be smooth varieties and the complement of A G in Y G is a divisor with normal crossings, described in terms of G-nested sets. Moreover, their integer cohomology rings are torsion free (see [3]). In [8] Yuzvinski eplicitly described Z-bases of these rings (see also [0]): we briefly recall these results concerning cohomology. Let G be a building set of subspaces of V. Let H G, let B G such that A B for each A H and define ( ) d H,B := dimb dim A. A H With these notations, in the polynomial ring Z[c A ] A G, we put P H,B := A H c A ( C B c C ) dh,b and we denote by I the ideal generated by these polynomials as H and B vary. Theorem.. (see [3]). There is a surjective ring homomorphism φ : Z[c A ] A G H (Y G,Z) with kernel is I and such that φ(c A ) H (Y G,Z). 3

4 Definition.5. Let G be a building set of subspaces of V. A function f : G N is G-admissible (or simply admissible) if f = 0 or, if f 0, supp(f) is G-nested and for all A supp(f) one has f(a) < d supp(f)a,a where supp(f) A := {C supp(f) : C A}. Definition.6. A monomial m f = A G cf(a) A Z[c A ] A G is admissible if f is admissible. Theorem.. (see [8] and also [0]) The set B G of all admissible monomials corresponds to a Z-basis of H (Y G,Z). 3 Maimal Models of Reflection Arrangements of Classical Type We now focus on the cohomology rings of maimal De Concini-Procesi models for root arrangements of type A n, B n (C n ) and D n. 3. Type A n Let W be a comple vector space of dimension n and consider the arrangement given by hyperplanes H ij := {z i z j = 0} where z i (i =,,n) are the coordinates. The intersection of these hyperplanes is the line N = {z = = z n }: we consider the quotient V = W/N and the arrangement provided by the images of the hyperplanes H ij via the quotient map W π V. We can choose linear functionals f ij in V such that the zeroes of f ij form the hyperplane π(h ij ) and the set {f ij } is a root system of type A n. In V we consider the subspace arrangement A An given by the lines < f ij > and denote for brevity by C An its closure under the sum and by F An the set of irreducible subspaces in C An. Our purpose is to compute the series φ A (q,t) = t+ n P CAn (q) tn n! Q[q][[t]] () where, for every n, P CAn (q) = ( (rk H i( ))) Y CAn,Z q i is the Poincaré polynomial of Y CAn (the variable q has degree ). In [8], Yuzvinsky noticed there is a bijective correspondence (actually an isomorphism of partially ordered sets) between the elements of F An and the subsets of {,, n} of cardinality at least two: this correspondence identifies the subset {i,,i k } with < f ii,,f ik i k >. Since every subspace in C An has a unique decomposition into irreducible subspaces, we can identify 4

5 elements of C An with families of disjoint subsets of cardinality at least two of {,,n}. Furthermore, given two such collectionsx = {X,,X k } andy = {Y,,Y r } we say that Y is included in X (and write Y X) if for every i {,,r} there eists j {,,k} such that Y i X j ; our identification thus becomes an isomorphism of partially ordered sets (we order C An by inclusion). A C An -nested set is a subset of C An strictly ordered by inclusion. Now we will see how to associate a graph (actually a forest of levelled oriented rooted trees) to a C An -nested set. Let us first recall by an eample the F An case (see [8]). Let us take, as F A8 -nested set, the collection S := {(,,3,4,5),(6,7,8,9),(,4,5),(6,7)}; we associate to S the following graph: (,,3,4,5) (6,7,8,9) (,4,5) 3 (6,7) where the edges are directed from the top to the bottom. Let us now consider the C A5 -nested set S := {(,,3,4,5)(8,0,,3,4,6),(,,4,5)(8,0,),(,),(0,)}. We associate to it the levelled forest Γ(S): (,,3,4,5) (8,0,,3,4,6) (,,4,5) 3 (8,0,) (,) 4 5 (0,) 8 0 where, again, the orientation is from the top to the bottom, and the elements of the nested set can be read level by level from the vertices which are not leaves (we call level the level which contains the roots, and level k+ the one which contains the vertices which are k steps away from a root). Let now S be a C An -nested set and let us denote by B (resp. A) the element of S determined by the vertices (not leaves) at level k (resp. k + ). Then A is the maimal element of S strictly contained in B. Hence if B is given by the family {B,,B k }, A by {A,,A r } and, for every i {,,k}, we set I Bi := {j {,,r} : A j B i }, we have dimb i j I Bi dima j = out(v Bi ) where v Bi is the verte of Γ(S) associated to B i and out(v Bi ) is the set of outgoing edges from v Bi. Then we obtain: d {A},B = ( out(v) ) () v Lev(k) where Lev(k) is the set of vertices (not leaves) of Γ(S) belonging to level k. 5

6 Definition 3.. A levelled forest Γ is admissible if, for any of its level k, one has ( out(v) ) = out(v) Lev(k). v Lev(k) v Lev(k) Definition 3.. Let Γ be a levelled admissible forest on n leaves. We denote by cont(γ) the contribution given to the series () by all the monomials m f of the basis such that supp f is a nested set whose graph is (up to a relabelling of the leaves) isomorphic to Γ. Proposition 3.. Let Γ be a levelled admissible forest on n leaves. Then we have n! cont(γ) = Aut(Γ) C Γ(q) tn n! where C Γ (q) = k level q v Lev(k) ( out(v) ) q q and Aut(Γ) is the group of automorphisms of Γ. Proof. We notice that there are n! Aut(Γ) different C A n -nested sets whose associated graph is Γ. The thesis follows by observing that, if Γ = Γ(S) where S is a C An -nested set, the contribution to the Poincaré polynomial P CAn (q) of the monomials m f such that supp f = S is C Γ (q). Our idea to compute the series () is to consider all levelled forests (not necessarily admissible) on at least two leaves and associate, to each of them, a monomial that encodes data we are interested in: number of levels and, for each level k, the number v Lev(k) ( out(v) ). We will put together these monomials in a series, which will be calculated inductively, and from which one can obtain the series (). Definition 3.3. A trivial tail of a (levelled) oriented forest T is given by a subtree T which stems from a verte v of T with out(v) = and has a single leaf. Definition 3.4. Two (levelled) oriented forests T and T are equivalent if they differ only for trivial tails. Definition 3.5. Given an equivalence class of (levelled) rooted oriented forests modulo trivial tails we call minimal representative the tree in this class with no trivial tails. Let us now define the following series p A (g 0,g,g,g 3, ) := g 0 + Γ g tr(γ) 0 Aut(Γ) v g out(v) l(v) where Γ runs among minimal representatives of levelled oriented forests on at least two leaves, tr(γ) is the number of trees of Γ (we are considering also the degenerate tree given by a single leaf), v varies among the vertices (not leaves) of Γ and l(v) is the level of v. 6

7 Definition 3.6. A monomial of p A is bad if there eist i < j such that g j appears in the monomial but g i doesn t. We notice that bad monomials correspond to forests with a level k such that ( out(v) ) = 0. v Lev(k) Definition 3.7. A monomial m of p A has valency k if k = ma{j : g j appears in m}. Proposition 3.. Given a levelled oriented forest Γ on n leaves, let m T be the monomial of p A associated to T. Then the degree of m Γ is n. Proof. One can restrict to trees and then proceed by induction on the valency of m Γ. Theorem 3.. Removing bad monomials from p A and replacing g 0 with t and, for every i,r, g r i with qr q q tr we obtain the Poincaré series (). Proof. Let s start by observing that, if we remove bad monomials, we haven t removed all monomials corresponding to non admissible forests; in fact we still have the ones corresponding to forests in which there is (at least) a level, say k, such that ( out(v) ) =. v Lev(k) Anyhow the contribution of such monomials is killed by our substitution; indeed they have a variable whose eponent is and qr q q = 0 if r =. Let now Γ be an admissible forest on n leaves; let m Γ be the monomial of Γ in p A and k be its valency. For all j k the eponent of the variable g j in m Γ is ( out(v) ). v Lev(j) Our claim then follows from Proposition 3. and Proposition 3.. The problem is now reduced to the computation of p A. To this end we define p A := + T Aut(T ) v g out(v) l(v) where T runs among minimal representatives of levelled oriented trees on n leaves and v among the vertices (not leaves) of T. One immediately checks that: therefore, all we need is a formula for p A. Theorem 3.. The following recursive formula holds: p A = e g0 p A, (3) pa = eg p A [] g (4) where p A [] is p A (g,g,g 3, ) evaluated in (g,g 3,g 4, ). 7

8 Proof. The formula is recursive, as one can easily check by induction on the valency. Let T be a tree on n leaves (recall that we are taking into account only minimal representatives modulo trivial tails). Let i m i mr r be a partition of n of lengthkmade by positive integersi,,i r such that, for eachj {,,r}, i j occurs m j times (and k = r j= m j). Suppose that eactly k edges stem from the root; furthermore, suppose that, if we cut off the root of T and these edges, we get a forest of k trees, {T,,T k }, such that, for each j {,,r}, m j of them are isomorphic and have i j leaves (here we are considering also the degenerate tree given by a single leaf). If, for every i {,,k}, we call m Ti the monomial of T i in p A and m T the one of T we have: m T = g k m!m! m r! k m Ti []. We conclude by observing that k i= m k! T i [] appears eactly ) k. in ( p A [] i= m!m! m r! times 3. Some Eamples Theorem 3. allows us to compute p A ; once we have p A we can compute p A and, using theorem 3., the series φ A ; here we ehibit some eamples of these computations made with the help of the Computer Algebra system Aiom. As a first eample, we show the monomials of p A of valency less than or equal to 3 and degree less than or equal to 3: ( 7 4 g g g + ) ( ( g g + 4 g + ) g + 4 g + g + ) g 3+ + ( 7 4 g3 + 4 g + ) ( g g + g + ) g + 4 g3 + 6 g + g +. If, for eample, we look at terms of degree 3 we have g3 3 4! + g g3 + g g3 + g g 3 + g g3 + g g3 + g g g 3 + g g g 3 + g g The nine monomials correspond to the levelled trees on 4 leaves with 3 levels (modulo equivalence): 8

9 Then we show p up to degree 5, without bad monomials: Degrees, and 3 g 0 + g 0 g + g 0 + g 0 g g + 6 g 0 g + g 0 g + 6 g3 0 Degree g 0 g g g g 0 g g g 0 g + 4 g3 0 g + 4 g4 0 ( 4 g 0 g + 3 ) 4 g 0 g Degree 5 3 g 0 g g g 3 g ( 7 8 g 0 g g g3 + g 0 g g + + ( 5 8 g 0 g g g 0 g + 5 ) 8 g 0 g g g 0 g g 0 g g3 0 g + g4 0 g + 0 g5 0 g + 4 g 0 g 3 + ( g 0 g + 3 ) g 0 g ( g 0 g g 0 g + g3 0 g At last here it is the series φ A up to degree 7 (with respect to t): φ A (q,t) = t+ ( t + 6 q + ) ( t q + 3 q + ) t ( + 0 q q q + ) t ( + 70 q q q q + ) t ( q5 + 9 q q q + 9 q + ) t ) g g 3 + ) g Type B n Let V be a comple vector space of dimension n. We consider in V the line arrangement corresponding to a root system of type B n and denote by C Bn its closure under the sum and by F Bn the set of irreducible subspaces of C Bn. Our aim is to compute the series: φ B (q,t) := t + n P CBn (q) tn n n! Q[q][[t]] (5) 9

10 where, for each n, P CBn (q) is the Poincaré polynomial of Y CBn. In [8], Yuzvinsky divided the elements of F Bn in two classes: strong elements and weak elements; if we call,, n V the coordinate functions, strong elements are the subspaces of V like < i,, ik > (k ), whose annihilator in V is the subspace H i,,i k := { i = = ik = 0}. They can be put in bijective correspondence with subsets of {,,n} of cardinality greater then or equal to (such subsets will be called strong). A weak element is a subspace whose annihilator is of type L i,,i k,j,...,j s := { i = = ik = j = = js } (r + s ); therefore weak elements can be put in a bijective correspondence with subsets of {,,n} of cardinality greater than or equal to equipped with a partition (possibly trivial) into parts (such subsets will be called weak). Moreover, if we order F Bn by inclusion of subspaces, we can read this order as follows: a subset that includes a strong subset of {,,n} is strong; a weak subset A is smaller than a strong subset B if and only if A B; a weak subset A = A A is smaller than a weak subset B = B B if and only if either A i B i (i =,) or A B and A B. Coming to the maimal building set, we observe that there is a bijective correspondence between elements of C Bn and families of disjoint subsets of {,,n} in which at most one is strong and in each of the weak ones a partition into two parts is fied. Given two such families X = {X,,X k } and Y := {Y,,Y h } we say that X is greater than Y (and write X Y ) if for every i {,,h} there eists j {,,k} such that Y i X j as elements of F Bn. It is again possible to associate levelled forests to C Bn -nested sets (a C Bn - nested is a subset of C Bn strictly ordered by inclusion); the rules are the same as in the case A n but we have to divide the vertices of our graphs into two classes: weak vertices and strong vertices. We notice that we lose the information concerning partitions of weak sets. From now we call "strong tree" a tree with at least one strong verte and "weak tree" a tree with no strong vertices; a forest is strong if it contains a strong tree, otherwise is weak. Let now S be a C Bn -nested set and Γ(S) be its associated forest; let us denote by B (resp. A) the element of S determined by the vertices (not leaves) at level k (resp. k + ); then A is the maimal element of S strictly contained in B. If B is given by a family of weak subsets, d {A},B can be computed, in terms of outgoing edges, eactly as in the A n case. Otherwise, B is associated to a family {B,,B k } (k ) of subsets of {,,n}, where B is strong. Then we have k d {A},B = wout(v B ) + ( out(v Bi ) ) (6) where v Bi is the verte of Γ(S) which corresponds to B i and wout(v B ) is the set of outgoing edges from v B to a weak verte (we are considering the leaves as weak vertices). The following lemma and corollary eplain how to take in account the information on the partitions associated to weak sets, which is not contained in the graphs. i= 0

11 Lemma 3.. (see [8]). Let S be a F Bn -nested set and Γ(S) be its associated forest. If we denote by π(γ(s)) the number of different F Bn -nested sets U such that Γ(U) = Γ(S), then log π(γ(s)) = v B dimb where v B ranges over all the maimal weak vertices (not leaves), i.e. the weak vertices (not leaves) which are not preceded, according to the orientation, by other weak vertices). Corollary 3.. Let Γ = Γ(S) be a levelled forest associated to a C Bn -nested set S. Let {v X,,v Xj } be the maimal weak vertices of Γ(S). Then Γ corresponds to j i= ( Xi ) different C Bn -nested sets. As in the A n case, to compute the Poincaré series (5) we define a series in infinite variables g 0,g,g,. We need to etend the definition of trivial tail to strong trees. Definition 3.8. A trivial tail of a levelled oriented strong tree T is given by a weak subtree T on a single leaf which stems from a verte v of T with out(v) =. Then we define a series which will take into account the contribution of strong forests to the Poincaré series: Q B(g 0,g,g, ) := Γ Aut(Γ) ( g0 ) tr(γ) v Γ s ( gl(v) ) wout(v) g out(v) l(v) v Γ w where Γ runs among minimal representatives (modulo trivial tails) of levelled oriented strong forests on n leaves, Γ s is the set of strong vertices of Γ, Γ w is the set of weak vertices (not leaves); we notice that an automorphism sends strong vertices to strong vertices and weak vertices to weak vertices. Then we put: Q B (g 0,g,g, ) := p A ( g 0,g,g, )+ Q B (g 0,g,g, ). (7) Remark 3.. We have that p A ( g0,g,g, ) = e g 0 p A (g,g, ). We notice that, if Γ is a (strong or weak) levelled oriented forest on n leaves then the corresponding monomial m Γ in Q B (g 0,g,g, ) has degree n. Theorem 3.3. If we remove bad monomials from Q B and replace g 0 with t and, for every i,r, g r i with qr q q tr, we obtain the series (5). Proof. It is a computation very similar to the one of Theorem 3.: p A ( g0,g,g, ) counts the contribution of weak forests, Q B of strong forests. We now need to compute Q B ; to this end we define Q B (g,g, ) := T Aut(T ) v T s ( gl(v) ) wout(v) g out(v) l(v) v T w

12 where T runs among minimal representatives of classes of strong trees on n leaves, T s is the set of strong vertices of T and T w is the set of the weak ones (not leaves). Then, since each strong forest has eactly one strong tree we have: Q B = ( p A ( g ) 0,g,g, )+ QB. Theorem 3.4. The following inductive formula holds: Q B = p A ( g 0,g,g, )[] + QB [] Proof. We will prove the equivalent formula ( ) j g j p A (g,g, )[] Q B = QB []+ j! j j 0 ( + p A ( g ) 0,g,g, )[]. (8) g j ( ) j p A (g,g, )[] j!. (9) Let T be a strong tree on n leaves; suppose that T has only one strong verte (therefore its root is strong). Let i m i mr r be a partition of n of length k made by positive integers i,,i r such that, for each j {,,r}, i j occurs m j times. Let us suppose that k edges stem from the (strong) root of T and that, if we cut off the root of T and these edges, we get a forest of k weak trees, {T,,T k }, such that, for each j {,,r}, m j of them are isomorphic and have i j leaves. If we call m T the monomial of T in QB and, for every i {,,k}, m Ti the one of T i in p A (g,g, ), we have m T = g k m!m! m r! k k m Ti []. To obtain the second addendum on the right side of formula (9) it s enough to notice that k k. i= m k! T i [] appears eactly m times in!m! m r! ( p A (g,g, )[]) Suppose now that T has more than one strong verte and that the (strong) root of T is connected to j + vertices such that j of them are weak and one (which we will denote by v) is strong. If j = 0 then eactly one edge stems from the root of T and (by assumption) it reaches v. If we call T the tree which stems from v we have that m T = m T []. Let j > 0 and suppose that the j subtrees whose roots are the j weak vertices are divided into h subsets containing respectively m,m,...,m h isomorphic trees. Let us denote by {T,,T j } these trees and let T j+ be the strong tree whose root is v. We have m T = i= g j m!m! m h! jm T j+ [] j m Tq [] where, as usual, m T is the monomial of T in QB, m Tj+ is the one of T j+ and, q= for each q {,,h}, m Tq is the monomial of T q in p A. We end by observing that k i= m j! T i [] appears eactly m!m! m h! times in p j A.

13 3.4 Type D n Let V be a comple vector space of dimension n. We consider in V the line arrangement corresponding to a root system of type D n and denote by C Dn its closure under the sum. The series we are interested in is the following: φ D (q,t) := t+q t 4 + n 3P CDn (q) Q[q][[t]] (0) n!n where, as usual, for each n 3, P CDn (q) is the Poincaré polynomial of Y CDn. The combinatorics is essentially the same as in the caseb n : the only difference is that strong sets must have cardinality at least two and then, for the computation of the Poincaré series, we just need to modify a little what we have done in that case. We set Q D(g 0,g,g, ) := Γ Aut(Γ) t n ( g0 ) tr(γ) v Γ s ( gl(v) ) wout(v) g out(v) l(v) v Γ w where Γ is a strong levelled oriented forest whose strong vertices correspond to subsets of cardinality least two and Q D (g,g, ) := T Aut(T ) v T s ( gl(v) ) wout(v) g out(v) l(v) v T w where the strong vertices of the strong levelled oriented tree T correspond to subsets of cardinality least two. Now, if we define Q D := p A ( g0,g,g, )+ Q D we can compute the Poincaré series in the same way described in theorem 3.3; moreover we have that j 0 Q D = ( p A ( g ) 0,g,g, )+ QD and QD satisfies a recurrence relation similar to (9): ( ) j ( g j p A (g,g, )[] Q D = QD []+ g j p A (g,g, )[] j! j! j ) j g. 4 Induced subspace arrangements The tensor product allows us to obtain new building subspace arrangements G h starting from a given building arrangement G in V. Definition 4.. We will call induced by G the subspace arrangement G h in V C h (h ) given by the subspaces A C h, as A varies in G. For instance, if G is a building set associated to a root system of type A, the complements of the arrangements G h are classical generalizations of the pure braid space (see [] and [6]). 3

14 It is immediate to check that, for any given building arrangement G in V, G h is still building, and therefore one can consider its De Concini-Procesi model. Let us focus on the case when the starting arrangements are the maimal building sets of type A,B(= C),D. Our series p A,Q B,Q D allow us to obtain quickly the Poincaré series of the families of models associated to the induced building sets: we only have to perform different substitutions for the variables g 0,g,... For instance, let us fi h and consider the A case: after removing bad monomials from p A, if we replace g 0 with t as before and, for every i,r, gi r with qrh q q tr, we obtain the Poincaré series for the models Y (CAn ) h (the same substitutions work also in the other cases). References [] Cohen, F., Lada, T., and May, J. The homology of iterated loop spaces. Springer, 976. [] De Concini, C., and Procesi, C. Hyperplane arrangements and holonomy equations. Selecta Mathematica (995), [3] De Concini, C., and Procesi, C. Wonderful models of subspace arrangements. Selecta Mathematica (995), [4] De Concini, C., and Procesi, C. On the geometry of toric arrangements. Transform. Groups 0 (005), [5] De Concini, C., and Procesi, C. Topics in Hyperplane Arrangements, Polytopes and Bo-Splines. Springer, Universitet, 00. [6] Drinfeld, V. On quasi triangular quasi-hopf algebras and a group closely connected with Gal(Q/Q). Leningrad Math. J. (99), [7] Etingof, P., Henriques, A., Kamnitzer, J., and Rains, E. The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. Annals of Math. 7 (00), [8] Feichtner, E. De Concini-Procesi arrangement models - a discrete geometer s point of view. Combinatorial and Computational Geometry, J.E. Goodman, J. Pach, E. Welzl, eds; MSRI Publications 5, Cambridge University Press (005), [9] Feichtner, E., and Kozlov, D. A desingularization of real differentiable actions of finite groups. Int. Math. Res. Not. (005), [0] Gaiffi, G. Blow ups and cohomology bases for De Concini-Procesi models of subspace arrangements. Selecta Mathematica 3 (997), [] Gaiffi, G. Compactification of configuration spaces. Seminari Algebra e Geometria, Scuola Normale Superiore (999). [] Henderson, A. Rational cohomology of the real coeter toric variety of type a. arxiv: (00). 4

15 [3] Lambrechts, P., Turchin, V., and Volic, I. Associahedron, cyclohedron, and permutohedron as compatifications of configuration spaces. Bull. Belg. Math. Soc. Simon Stevin 7 (00), [4] Postnikov, A. Permutohedra, associahedra, and beyond. Int Math Res Notices (009), [5] Postnikov, A., Reiner, V., and Williams, L. Faces of generalized permutohedra. Documenta Mathematica 3 (008), [6] Sundaram, S., and Welker, V. Group actions on arrangements of linear subspaces and applications to configuration spaces. Trans. Amer. Math. Soc 349 (997), [7] Szenes, A., and Vergne, M. Toric reduction and a conjecture of Batyrev and Materov. Invent. Math. 58 (004), [8] Yuzvinsky, S. Cohomology bases for De Concini-Procesi models of hyperplane arrangements and sums over trees. Invent. Math. 7 (997), [9] Zelevinski, A. Nested complees and their polyhedral realizations. Pure and Applied Mathematics Quarterly (006),