Colored partitions and a generalization of the braid arrangement

Volkmar Welker^{1}

Fachbereich 6, Mathematik

Universität GH-Essen

D-45117 Essen, Germany

                   welker@exp-math.uni-essen.de

Submitted: November 11, 1996; Accepted: November 22, 1996.

Abstract

We study the topology and combinatorics of an arrangement of hyperplanes in C^{n} that generalizes the classical braid arrangement. The arrangement plays in important role in the work of Schechtman & Varchenko [12, Part II] on Lie algebra homology, where it appears in a generic fiber of a projection of the braid arrangement. The study of the intersection lattice of the arrangement leads to the definition of lattices of colored partitions. A detailed combinatorial analysis then provides algebro-geometric and topological properties of the complement of the arrangement. Using results on the character of S_{n} on the cohomology of these arrangements we are able to deduce the rational cohomology of certain spaces of polynomials in the complement of the standard discriminant that have no root in the first s integers.

1 Introduction

In this paper we study the arrangement A_{n}^{col,s} of all affine hyperplanes H_{ij} : z_{i} = z_{j}, 1 ≤ i < j ≤ n, and H_{i}^{r} : z_{i} = r, 1 ≤ i ≤ n and 1 ≤ r ≤ s. This arrangement appears in the work of Schechtman & Varchenko [12, Part II] as a generic fiber of projections of the braid space in the context of Lie-algebra homology. We investigate the combinatorics of the intersection lattice L_{A_{n}^{col,s}} of A_{n}^{col,s} (i.e., the set of all subspaces that are intersections of hyperplanes in the arrangement, ordered by reversed inclusion). This leads to the definition of “colored partitions.” Via the analysis of the homology of the order complex of the intersection lattice and using a formula by Orlik & Solomon [10] we are able to determine the cohomology of the complement C^{n} \ ⋃_{H∈A_{n}^{col,s}} H. The sym-

1Supported by the DFG through “Habiliationsstipendium” We 1479/3 Keywords: Partition, hyperplane arrangement, intersection lattice, configuration space Mathematics Subject Classification. Primary 05C40, 52B30. Secondary 05C40, 05E25

metric group S_{n} acts on C^{n} by permuting the coordinates and leaving A_{n}^{col,s} invariant. By a calculation of the character of S_{n} on the homology of the order complex of L_{A_{n}^{col,s}} and using a formula of Orlik & Solomon [10] we are able to describe the character of S_{n} on the cohomology of the complement C^{n} \ ⋃_{H∈A_{n}^{col,s}} H. Passing to rational coho-

mology and computing the space of S_{n}-invariants on the cohomology allows then a description of the rational cohomology of the quotient (C^{n} \ ⋃_{H∈A_{n}^{col,s}} H)/S_{n}. The latter

can then be identified with the space of polynomials f(X) = X^{n} +a_{n−1}X^{n−1} +⋯+a_{0} in C^{n} such that f(X) has no double root and no root of f(X) lies in [s] := {1, …, s}. These spaces can be used to approximate the space of monic polynomials of degree n that have no double and no integral root.

2 Basic Definitions

An arrangement A of hyperplanes in C^{n} is a finite set of affine hyperplanes in complex n-space. To each arrangement A corresponds an n-dimensional complex manifold M_{A} = C^{n} \ ⋃_{H∈A} H. The space M_{A} is called the complement of the arrangement

A. The combinatorial object associated to an arrangement A is the intersection (semi)lattice L_{A}. It is the set of subspaces V of C^{n} such that ∅ ≠ V = ⋂_{H∈B} H for

some subset B ⊆ A ordered by reversed inclusion. Here we allow B = ∅ and identify the intersection ⋂_{H∈∅} H with the space C^{n}. Note, that in general L_{A} is actually not

a lattice but a meet-semilattice (i.e., infima exist but suprema in general not). The link between the combinatorics of L_{A} and the topology of M_{A} is provided by the order complex of lower intervals in L_{A}. In general, for a partially ordered set P with top element ˆ1 and least element ˆ0 we denote by Δ(P) the order complex of P. This is the simplicial complex whose simplices are the chains x_{0} < ⋯ < x_{l} in P \ {ˆ0, ˆ1}. For x ≤ y, x, y ∈ P, we write [x, y] to denote the interval {z | x ≤ z ≤ y} in P. If a finite subgroup G ≤ Gl_{n}(C) acts on C^{n} leaving M_{A} invariant then G also acts on L_{A} as a group of lattice automorphisms. If V ∈ L_{A} then the stabilizer Stab_{G}(V) = {g ∈ G | V^{g} = V } of V in G acts on the lower interval [C, V] in L_{A} as a group of lattice automorphisms. These actions induce a representation of G on the cohomology of M_{A} and a representation of Stab_{G}(V) on the homology of the order complex Δ([C, V]). The following result by Orlik & Solomon [10] links these two representations.

Proposition 2.1 [10] Let G ≤ Gl_{n}(C) be a finite group and let A be an arrangement of affine hyperplanes in C^{n} such that M_{A} is invariant under G. Then,

H̃^{i}(M_{A}) ≅ ⨁_{V ∈L_{A}/G\{C}} ind_{Stab_{G}(V)}^{G}H̃_{codim(V)−i−2}(Δ([C, V])),

where L_{A}/G is a set of representatives of G-orbits on the lattice L_{A}.

A result by Ziegler & Živaljević [21] is concerned with the union U_{A} = ⋃_{H∈A} H of

an arrangement A. The result by Ziegler & Živaljević is actually far more general and is valid for general arrangements of linear subspaces. Here, we state an equivariant version of the result by Ziegler & Živaljević that can be found in [19].

Proposition 2.2 [21] Let A be an arrangement of affine hyperplanes in C^{n}. Assume G ≤ Gl_{n}(C) is a finite subgroup that leaves A invariant. Let ˆ1 be an additional element that is larger than any element of L_{A}. Then, U_{A} is G-homotopy equivalent to Δ(L_{A} ∪ {ˆ1}).

Based on these results, we start the investigation of the special class of arrangements we want to consider in this paper. Before we proceed, we recall a general method to determine the homotopy type of the order complex of a poset P. The formula is due to Björner & Walker [3] for G = 1 and can be found in [18] in the general case.

Lemma 2.3 [3] (Homotopy Complementation Formula) Let L be a (finite) lattice with least element ˆ0 and largest element ˆ1. Assume G is a finite group of automorphisms of L. Let a ∈ L\{ˆ0, ˆ1} be a G-invariant element. Denote by Co(a) = {x ∈ L | inf(x, a) = ˆ0, sup(x, a) = ˆ1} the set of complements of a. Then Δ(L) is G-homotopy equivalent to the wedge

⋁_{x∈Co(a)} susp(Δ([ˆ0, x]) ∗ Δ([x, ˆ1])),

where G permutes the spaces in the wedge according to the action of G on L.

In the formulation of the lemma we denote by “inf” the infimum operation in L and by “sup” the supremum operation in L. We write “⋁” for the wedge of topological spaces. Recall, that the wedge X ∨ Y of two topological is the disjoint union of X and Y modulo the identification of one point x ∈ X with one point y ∈ Y. Note, that without specifying the points the wedge is (modulo homotopy) well defined whenever all spaces are path-connected. It turns out, that this is the case in the formula given by Lemma 2.3, except for some discrete 2-point spaces, where the wedge point has to be chosen to be one of the points. By “susp” we denote the suspension operation and by “∗” we denote the join operation. Note, that in contrast to the common usage, we define the join of a space X with the empty set to be the space X itself and not the empty set. For more detailed information and the definitions we refer the reader to Munkres’ book [9].

3 A Generalization of the Braid Arrangement

The classical braid arrangement A_{n} in complex n-space is given by the “thick” diagonals H_{ij} : z_{i} = z_{j} for 1 ≤ i < j ≤ n. The braid arrangement, also known as the complexified Coxeter arrangement of type A, is a well studied object (see for example Fox & Neuwirth [7], Arnol’d [1], Brieskorn [4] and Lehrer & Solomon [8]). Its name is derived from the fact that by a result of Fox & Neuwirth [7] the complement M_{A_{n}} is the classifying space of the pure braid group on n strings. We enlarge the central (i.e., all hyperplanes pass through the origin) arrangement A_{n} by some affine subspaces. Let A_{n}^{col,s} be the arrangement of complex hyperplanes H_{ij} : z_{i} = z_{j}, 1 ≤ i < j ≤ n and H_{i}^{r} : z_{i} = r for 1 ≤ i ≤ n, 1 ≤ r ≤ s. This arrangement occurs in the work of Schechtman & Varchenko [12, Part II]. More generally, Schechtman & Varchenko [12] consider the projection of the complement M_{A_{n+s}} of the braid arrangement A_{n+s} of hyperplanes H_{ij} : z_{i} = z_{j}, 1 ≤ i < j ≤ n + s in complex (n + s)-space on the last s coordinates. Let pr_{n,s} be the projection of (n + s)-space on the last s coordinates. The image of pr_{n,s} is complex s-space. For a point (t_{1}, …, t_{s}) in pr_{n,s}(M_{A_{n+s}}) (i.e., it satisfies t_{i} ≠ t_{j} for 1 ≤ i < j ≤ s) the fiber pr_{n,s}^{−1}(t_{1}, …, t_{s}) of pr_{n,s} when restricted to M_{A_{n+s}} is homeomorphic to M_{A_{n}^{col,s}}.

Let us define some combinatorial objects that turn out to be important in the investigation of the arrangement A_{n}^{col,s}. We describe a partition τ of the set [n] by B_{1}| ⋯ |B_{f}, where B_{i} ⊆ [n], B_{i} ∩ B_{j} = ∅, ⋃_{i=1}^{f} B_{i} = [n]. The sets B_{i} are called the blocks of τ. We denote by Π_{n} the lattice of all partition of [n] ordered by refinement (i.e., we say B_{1}| … |B_{f} ≤ C_{1}| … |C_{e} if f ≤ e and for each 1 ≤ i ≤ f there is a 1 ≤ j ≤ e such that B_{i} ⊆ C_{j}). Let Π_{n}^{col,s}, s ≥ 1, be the set of all pairs (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) of partitions τ ∈ Π_{n} and sequences of numbers l_{i} ∈ {0, …, s} of length t, where t is the number of blocks of τ and for each j ∈ [s] – note that then j ≠ 0 – there is at most one index i for which l_{i} = j. We say (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) is smaller than (τ′ = B′_{1}| ⋯ |B′_{t′}, (l′_{1}, …, l′_{t′})) if and only if τ ≤ τ′ and if B_{i} ⊆ B′_{j} then l_{i} ≠ 0 implies l′_{j} = l_{i}. We call an element (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) of Π_{n}^{col,s} a colored partition of [n]. We call the number l_{i} the color of the ith block of τ. The number “0” in this context stands for “no color.” If s = 1 then Π_{n}^{col,s} is actually a lattice with top element (|1 ⋯ n|, (1)). In general, let (τ, (l_{1}, …, l_{t})) and (τ′, (l′_{1}, …, l′_{t′})) be two colored partitions. Assume (γ, (m_{1}, …, m_{q})) is an upper bound of (τ, (l_{1}, …, l_{t})) and (τ′, (l′_{1}, …, l′_{t′})). Then if B_{i} is a block of τ (resp., τ′) then for the block C_{j} of γ that contains B_{i} we have m_{j} = 0 implies l_{i} = 0 (resp., l′_{i} = 0). Hence, we may assume that if B_{i} ⊆ C_{j} then l_{i} = m_{j} (resp., l′_{i} = m_{j}). We set τ′′ = τ ∨τ′ and for a block D_{j} of τ′′ we set n_{j} = l_{i} for any block B_{i} of τ contained in D_{j}. Then (τ′′, (n_{1}, …, n_{q′})) is an upper bound of (τ, (l_{1}, …, l_{t})) and (τ′, (l′_{1}, …, l′_{t′})) that is smaller than (γ, (m_{1}, …, m_{q})). Thus (τ′′, (n_{1}, …, n_{q′})) is the supremum of (τ, (l_{1}, …, l_{t})) and (τ′, (l′_{1}, …, l′_{t′})). In particular, this implies that all lower intervals in Π_{n}^{col,s} are lattices.

Proposition 3.1 Let 1 ≤ s, n. The intersection lattice L_{A_{n}^{col,s}} is isomorphic to the partially ordered set of colored partitions Π_{n}^{col,s}.

Proof. Let (τ = (B_{1}| … |B_{t}), (l_{1}, …, l_{t})) be a colored partition in Π_{n}^{col,s}. Then we map (τ = (B_{1}| … |B_{t}), (l_{1}, …, l_{t})) to the affine subspace V_{(τ,(l_{1},...,l_{t}))} that is defined by z_{i} = z_{j} if i and j lie in the same block of τ and z_{i} = l_{j} for i ∈ B_{j} in case l_{j} ≠ 0. Obviously, this is an order preserving map to L_{A_{n}^{col,s}}. Conversely, we map each element V ∈ L_{A_{n}^{col,s}} to the colored partition (τ = B_{1}| … |B_{t}, (l_{1}, …, l_{t})) that is defined by : i, j lie in the same block of τ if z_{i} = z_{j} and l_{j} = z_{i} if i ∈ B_{j} and z_{i} ∈ [s]. Obviously, the two maps are inverse to each other. One checks, that they induce indeed a poset isomorphism L_{A_{n}^{col,s}} ≅ Π_{n}^{col,s}.

A geometric semilattice is (see for example Wachs & Walker [17]) a meet-semilattice L that is constructed from a geometric lattice L′ by removing an upper interval [x, ˆ1] for an atom x of L′ (i.e., L = L′ \ [x, ˆ1]). If L is a geometric semilattice then for each x ∈ L the number of elements in a maximal chain from the least element ˆ0 to x is independent of the choice of the maximal chain. We denote by rank(x) the rank of x in L (i.e., the number of elements in a maximal chain in [ˆ0, x] minus 1). As an immediate consequence we obtain :

Corollary 3.2 The partially ordered set Π_{n}^{col,s} is a geometric semilattice. In particular, if ˆ1 is an additional element and s > 1 then the order complex Δ(Π_{n}^{col,s} ∪ {ˆ1}) is homotopic to a wedge of spheres of dimension n − 1. For s = 1 the complex Δ(Π_{n}^{col,s} ∪ {ˆ1}) is contractible. More generally, for an element x ∈ Π_{n}^{col,s} the order complex of interval [ˆ0, x] is homotopic to a wedge of spheres of dimension rank(x)−2.

Proof. It is well known that the intersection lattice of an affine hyperplane arrangement is a geometric semilattice (see for example [11]). The corresponding geometric lattice can be constructed by enlarging the arrangement by a hyperplane at infinity and then considering the intersection lattice of the enlarged arrangement. By a result of Wachs & Walker [17] the order complex Δ(L ∪ {ˆ1}) of a geometric semilattice L enhanced by an additional top element is homotopic to a wedge of spheres of dimensions rank(L ∪ {ˆ1}) − 2. Also, for x ∈ L the order complex of the interval [ˆ0, x] is homotopic to a wedge of spheres of dimension rank(x) − 2. Then the result for s > 1 and for intervals follows from Proposition 3.1. It remains to treat the case s = 1. As mentioned before in this case the (semi)lattice Π_{n}^{col,s} has a top element (see also Remark 3.3 and Proposition 4.2). Thus the order complex of Π_{n}^{col,s} ∪ {ˆ1} is a cone and hence contractible.

In order to give the reader a feeling for the combinatorial structure of the lattice of colored partitions, we classify the cover relations in Π_{n}^{col,s}. Let

(τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) < (τ′ = B′_{1}| ⋯ |B′_{t′}, (l′_{1}, …, l′_{t′}))

be a cover relation in Π_{n}^{col,s}. Then either :

(A) τ = τ′, there is a unique index i such that l_{j} = l′_{j} for j ≠ i and l_{i} = 0, l′_{i} ≠ 0.

(B) τ < τ′ in Π_{n} is a cover relation and τ′ is constructed from τ by merging the blocks B_{i} and B_{h} into the block B′_{j} for which l_{i} = l_{h} = l′_{j} = 0.

(C) τ < τ′ in Π_{n} is a cover relation and τ′ is constructed from τ by merging the blocks B_{i} and B_{h} into the block B′_{j} for which l_{i} = l′_{j} ≠ 0 and l_{h} = 0.

The following Remark 3.3 was first stated implicitly by Edelman & Reiner [5]. They made the observation on the realm of arrangements that are extensions of the braid arrangement by some set of hyperplanes defined by equations z_{i} = ±z_{j} and z_{i} = 0. Of course, this includes the arrangement A_{n}^{col,1}. The case s > 1 is not considered by Edelman & Reiner, their motivation for studying the corresponding arrangements origins in the “freeness” condition (see the book by Orlik & Terao [11]) and therefore there is no further overlap with the work presented here.

Remark 3.3 The lattice Π_{n}^{col,1} is S_{n}-isomorphic to Π_{n+1}.

Proof. We map a colored partition of [n] to the partition of [n + 1] that is defined by adjoining n+1 to the colored block, in case there is one; or adjoining the singleton |n+ 1| in case there is no colored block. It is easily seen that this defines an S_{n}-equivariant (S_{n} regarded as the subgroup of S_{n+1} stabilizing n + 1) lattice isomorphism.

4 Combinatorics & Homology of Lattices of Colored Partitions

In this section we determine the G-homotopy type of the posets [ˆ0, (τ, (l_{1}, …, l_{t}))] where G is the stabilizer of (τ, (l_{1}, …, l_{t})) in S_{n}. First, we consider the structure of intervals [ˆ0, (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t}))]. After possibly renumbering the blocks we may assume that l_{f} = ⋯ = l_{t} = 0 and l_{1}, …, l_{f−1} ≠ 0.

Lemma 4.1 Let G be the stabilizer of the colored partition (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) in S_{n}. Assume that l_{f} = ⋯ = l_{t} = 0 and l_{1}, …, l_{f−1} ≠ 0. The interval [ˆ0, (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t}))] is G-isomorphic to

×_{i=1}^{f−1}Π_{|B_{i}|}^{col,1} × ×_{i=f}^{t}Π_{|B_{i}|} ≅ ×_{i=1}^{f−1}Π_{|B_{i}|+1} × ×_{i=f}^{t}Π_{|B_{i}|}.

Proof. The isomorphism to the poset on the left hand side is obvious, since all blocks can be split independently. The second isomorphism then follows from Remark 3.3.

By the previous lemma it suffices to consider the S_{n}-lattices Π_{n}^{col,1} in order to understand the G-homotopy type of lower intervals in Π_{n}^{col,s}.

Proposition 4.2 The S_{n}-homotopy type of Π_{n}^{col,1} is given by a wedge of n! spheres of dimension (n − 2). The n! spheres are permuted by S_{n} according to its regular representation. In particular, H̃_{n−2}(Π_{n}^{col,1}) is the regular S_{n}-module.

Proof. Let (|1 ⋯ n|, (0)) be the maximal element in Π_{n}^{col,1} with no colored block If a colored partition (τ, (l_{1}, …, l_{t})) is a complement of (|1 ⋯ n|, (0)) then at least one (and therefore exactly one) index i must satisfy l_{i} = 1. Moreover, if there is a non-trivial block in τ then (τ, (0, …, 0)) is a lower bound for (|1 ⋯ n|, (0)) and (τ, (l_{1}, …, l_{t})). Thus any complement of (|1 ⋯ n|, (0)) must be of the form (|1| ⋯ |n|, (0, …, \underbrace{1}_{i}, … 0)) where the 1 is at the ith position. Hence there are n

complements of (|1 ⋯ n|, (0)) and they are permuted by S_{n} according to the natural S_{n}-action and each complement is stabilized by one of the one-point stabilizers S_{n−1} in S_{n}. Each complement is an atom in Π_{n}^{col,1} and the upper intervals [(|1| ⋯ |n|, (0, …, \underbrace{1}_{i}, … 0)), ˆ1] (ˆ1 being the largest element (|1 ⋯ n|, (1)) of Π_{n}^{col,1})

are S_{n−1}-isomorphic to Π_{n} ≅ Π_{n−1}^{col,1}. By the G-equivariant Homotopy Complementation Formula 2.3 the result follows.

Let us denote by r_{n} the character of the regular S_{n}-representation, by sgn_{n} the character of the sign-representation of S_{n} and by 1_{n} the character of the trivial S_{n}- representation. By π_{n} we denote the character of S_{n} on the homology of the order complex of Π_{n} in dimension n − 3. It is a well studied character of dimension (n − 1)! (see Stanley [13] for a detailed description).

Corollary 4.3 Let G be the stabilizer of (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) in S_{n}. Assume that l_{f} = ⋯ = l_{t} = 0 and l_{1}, …, l_{f−1} ≠ 0. Let B_{f}| ⋯ |B_{t} be a partition of type (1^{e_{1}}, …, n^{e_{n}}). Then

G ≅ S_{|B_{1}|} × ⋯ × S_{|B_{f−1}|} × S_{e_{1}}[S_{1}] × ⋯ × S_{e_{n}}[S_{n}].

The S_{n}-character on

ind_{G}^{S_{n}} H̃_{n−t+f−2}([ˆ0, (τ, (l_{1}, …, l_{t}))])

is given by

ind_{G}^{S_{n}}r_{|B_{1}|} ⋯ r_{|B_{f−1}|} · sgn_{e_{1}}[π_{1}] · 1_{e_{2}}[π_{2}] ⋯

Proof. The assertion follows immediately from Proposition 4.2 and the [16, Theorem 1.1].

We are grateful to Richard Stanley for pointing out that the characteristic polynomial (see [14]) of Π_{n}^{col,s} can be easily computed using a result about characteristic polynomials of hyperplane arrangements (see Orlik & Terao [11, Theorem 2.69]) or more generally subspace arrangements (Athanasiadis [2, Theorem 2.2]). The characteristic polynomial χ(P, t) of a poset P with rank function rank and minimal element ˆ0 is defined by

χ(P, t) = ∑_{x∈P} μ(ˆ0, x)t^{rank(P)−rank(x)}.

Here, rank(P) is the maximal rank of one of the elements of P and “μ” denotes the Möbius function of P (see [14]).

Proposition 4.4 Let A be an affine hyperplane arrangement in C^{n} such that the subspaces in A can be defined by equations using only integer coefficients. Let F_{q} denote the field with q elements, q a prime. By our assumption we then can regard A as an arrangement in F_{q}^{n}. Then for large enough q we have

χ(L_{A}, q) = ∥F_{q}^{n} \ ( ⋃_{H∈A} H)∥.

Corollary 4.5 The characteristic polynomial χ(Π_{n}^{col,s}, q) is given by

(q − s) ⋯ (q − s − n + 1).

Proof. If (x_{1}, …, x_{n}) is a point in the complement F_{q}^{n} \ ( ⋃_{H∈A} H) then if q is large

enough there are (q − s − (i − 1)) choices for the ith coordinate x_{i}. From this observation, the result follows from the preceding Proposition 4.4 and Proposition 3.1.

So far we have treated lower intervals in Π_{n}^{col,s}. Now we turn our interest to Π_{n}^{col,s} itself. Let us denote by ˆ1 an additional element that is larger than all elements of Π_{n}^{col,s}. Then by standard facts about the characteristic polynomial (see [14]) the preceding proposition immediately implies that μ(Π_{n}^{col,s} ∪ {ˆ1}) = χ(Π_{n}^{col,s}, 1) = (−1)^{n}(s − 1) ⋯ ((s − 1) + (n − 1)).

Proposition 4.6 The poset Π_{n}^{col,s} ∪ {ˆ1} is homotopy equivalent to a wedge of

(s − 1) ⋯ ((s − 1) + (n − 1))

spheres of dimension n − 1. The S_{n}-homotopy type of Π_{n}^{col,s} ∪ {ˆ1} is a wedge of n! copies of a wedge of \frac{(n+s−2)!}{(s−2)!n!} spheres of dimension n − 1, that are permuted according to the regular S_{n}-representation. In particular, if s = 1 then Π_{n}^{col,s} ∪ {ˆ1} is contractible. The representation of S_{n} on H̃_{n}(Π_{n}^{col,s}) is given by \frac{(n+s−2)!}{(s−2)!n!} copies of the regular representation of S_{n}.

Proof. We give the non-equivariant part of the assertion. The equivariant part of the assertion follows using Proposition 2.2 from Theorem 5.1 (ii). Note, that in the proof of Theorem 5.1 (ii) we use the non-equivariant part of this Proposition 4.6.

By results of Wachs & Walker [17] the order complex of a geometric semilattice L enlarged by an additional top element ˆ1 is homotopic to a wedge of spheres of dimension rank(L ∪ {ˆ1}) − 2. In particular, the homology of the order complex is free of rank equal to the number of spheres and concentrated in one dimension. Since the Möbius number of a poset equals by a result of P. Hall (see for example [14]) the alternating sum of ranks of homology groups of the order complex of P, the result follows from the previous observations about the Möbius number.

5 Geometry and Topology of the Arrangement

Using results on the combinatorics of A_{n}^{col,s} presented in the preceding section, we obtain:

Theorem 5.1 Let 1 ≤ s, n.

(i) There is an isomorphism of S_{n}-modules

H̃^{i}(C^{n} \ U_{A_{n}^{col,s}}) ≅ ⨁_{p∈Π_{n}^{col,s}/S_{n}\{ˆ0}} ind_{Stab_{Sn}(p)}^{S_{n}}H̃_{codim(V_{p})−i−2}(Δ(ˆ0, p)),

where V_{p} is the subspace in L_{A_{n}^{col,s}} corresponding to p ∈ Π_{n}^{col,s}. In particular, H̃^{∗}(C^{n} \ U_{A_{n}^{col,s}}) is free.

(ii) If s > 1 then U_{A_{n}^{col,s}} is S_{n}-homotopic to a wedge of n! copies of a wedge of \frac{(n+s−2)!}{(s−2)!n!} spheres of dimension n − 1, where the n! spaces are permuted according to the regular representation of S_{n}. In particular, the space U_{A_{n}^{col,s}}/S_{n} is homotopic to a wedge of \frac{(n+s−2)!}{(s−2)!n!} spheres of dimension n. If s = 1 then U_{A_{n}^{col,s}} is S_{n}-contractible.

Proof. Part (i) follows immediately from Proposition 3.1, Proposition 2.1.

The proof of part (ii) is more subtle. If s = 1 then the arrangement A_{n}^{col,s} is equivalent to a central arrangement (consider the point (1, …, 1) as the origin). The map sending all points in U_{A_{n}^{col,s}} to the origin defines an S_{n}-deformation retraction. In particular, U_{A_{n}^{col,s}} is S_{n}-contractible (it is a general well known fact that the union of a central arrangement is contractible). Now consider the case s > 1. By Proposition 2.2 we have U_{A_{n}^{col,s}} ≃_{S_{n}} Δ(Π_{n}^{col,s} ∪ {ˆ1}). Thus it suffices to consider the S_{n}-homotopy type of U_{A_{n}^{col,s}}. Let us regard R^{n} as the subspace of C^{n} defined by the equations Im(z_{i}) = 0, 1 ≤ i ≤ n. Then U_{A_{n}^{col,s}}^{R} := U_{A_{n}^{col,s}} ∩ R^{n} is an S_{n}-deformation retract of U_{A_{n}^{col,s}}. The homotopy is given by K : U_{A_{n}^{col,s}}×[0, 1] → U_{A_{n}^{col,s}} that sends ((x_{1}+iy_{1}, …, x_{n}+iy_{n}), t) to (x_{1}+ity_{1}, …, x_{n}+ity_{n}). Actually this is a well known general fact about arrangements and their complexifications (see [21]). Let U_{A_{n}}^{R} be the “real part” U_{A_{n}} ∩ R^{n} of union of the braid arrangement A_{n} = {H_{ij} : z_{i} = z_{j} | 1 ≤ i < j ≤ n}. Then U_{A_{n}}^{R} is a S_{n}- invariant subspace of U_{A_{n}^{col,s}}^{R}. Moreover, U_{A_{n}}^{R} is S_{n}-contractible. The map L : U_{A_{n}}^{R} × [0, 1] → U_{A_{n}}^{R} defined by L((z_{1}, …, z_{n}), t) = t·(z_{1}, …, z_{n}) defines a S_{n}-homotopy from id_{U_{An}^{R}} to the constant map from U_{A_{n}}^{R} to the origin \underline{0}. Thus the inclusion \{\underline{0}\} ↪ U_{A_{n}}^{R} induces an S_{n}-deformation retract to a one point space. Therefore, by standard arguments the map U_{A_{n}^{col,s}}^{R} → U_{A_{n}^{col,s}}^{R}/U_{A_{n}}^{R} defines a S_{n}-homotopy equivalence. Let X_{n} be the closed simplicial cone R × R_{+}^{n−1} ≅ {(x_{1}, …, x_{n}) ∈ R^{n} | x_{1} ≤ ⋯ ≤ x_{n}}. Then the space U_{A_{n}^{col,s}}^{R}/U_{A_{n}}^{R} is a wedge of n! copies of Y_{n} = (X_{n} ∩ U_{A_{n}^{col,s}}^{R})/(X_{n} ∩ U_{A_{n}}^{R}). The n! spaces are permuted freely according to the regular representation, the image of U_{A_{n}}^{R} serves as the wedge point. We already know by the non-equivariant part of Proposition 4.6 that U_{A_{n}^{col,s}}^{R} is homotopic to a wedge of (s − 1) ⋯ ((s − 1) + (n − 1)) spheres of dimension n and no point in Y_{n} is fixed by an element of S_{n}. From this it follows that Y_{n} is homotopic to \frac{1}{n!} · (s − 1) ⋯ ((s − 1) + (n − 1)) spheres of dimension n.

Note, that in part (i) the conclusion that the cohomology is free and in part (ii) the conclusion that the union is homotopic to a wedge of spheres is known to be true in general for hyperplane arrangements (see [11] and [21]).

As another immediate consequence we obtain a result on the cohomology of the complement of A_{n}^{col,s} in a rank one local system. We emphasize this otherwise standard application of the combinatorial methods here, since it fits in the framework of the considerations by Schechtman & Varchenko [12]. Let A be some arrangement of affine complex hyperplanes in C^{n}. Let ξ = (ξ_{H})_{H∈A} be some vector of complex numbers.

Then we denote by ω_{ξ} the differential form ∑_{H∈A}ξ_{H} · \frac{dH}{H}, where H is identified with a

linear form defining H.

Proposition 5.2 Let 1 ≤ s, n. Let ξ = (ξ_{H})_{H∈A_{n}^{col,s}} be some vector of complex numbers, such that for all V ∈ L_{A_{n}^{col,s}} \ {ˆ0} the sum ∑_{H≤V} ξ_{H} over all hyperplanes H con-

taining V does not vanish. Then the rank of the cohomology H^{i}(C^{n} \ U_{A_{n}^{col,s}}, L_{ω_{ξ}}) of C^{n} \ U_{A_{n}^{col,s}} with coefficients in the rank one local system L_{ω_{ξ}} defined by ω_{ξ} is given by (s − 1) ⋯ (s + n − 2) for i = n and 0 in all other dimensions.

Proof. By the work of Esnault, Viehweg & Schechtman [6] it follows that under the given assumptions the cohomology with coefficients in the rank one local system vanishes except in dimension n. By general facts or by the work of Yuzvinsky [20] we have ∑_{i≥0}(−1)^{i}rankH^{i}(C^{n} \ U_{A_{n}^{col,s}}, L_{ω_{ξ}}) = χ(L_{A_{n}^{col,s}}, 1). Hence, the assertion follows from Corollary 4.5.

Finally, we turn our interest to the quotient spaces (C^{n} \ U_{A_{n}^{col,s}})/S_{n}. We recall a basic fact about symmetric products of complex lines.

Proposition 5.3 Let the symmetric group S_{n} act on complex n-space C^{n} by permuting the coordinates. Then the map that sends an n-tuple (z_{1}, …, z_{n}) to the polynomial f(X) = (X − z_{1}) ⋯ (X − z_{n}) induces an homeomorphism from C^{n}/S_{n} to C^{n}.

For Proposition 5.3 we immediately infer the following interpretation.

Lemma 5.4 The space (C^{n} \U_{A_{n}^{col,s}})/S_{n} is homeomorphic to the space of monic complex polynomials of degree n with no double root and no root in the set [s].

Using our description of the S_{n}-action on cohomology we obtain:

Theorem 5.5 Let 1 ≤ s, n.

(i) H̃^{1}((C^{n} \ U_{A_{n}^{col,s}})/S_{n}, Q) ≅ Q^{s+1}.

(ii) H̃^{i}((C^{n} \ U_{A_{n}^{col,s}})/S_{n}, Q) ≠ 0 for i = 2, …, n − 1. The rank is given by the number of S_{n}-orbits of elements (τ = B_{1}| … |B_{t}, (l_{1}, …, l_{t})) in Π_{n}^{col,s} such that l_{i} = 0 implies |B_{i}| = 1, 2 and there is at most one index i such that l_{i} = 0 implies |B_{i}| = 2.

Proof. In order to determine the rational cohomology of the quotient it suffices to determine the multiplicity of the trivial representation of S_{n} in the cohomology of C^{n} \ U_{A_{n}^{col,s}}. Note, that in general for a G-space X, which is CW-complex and G a finite group, the rational cohomology of X/G is given by the space of G-invariants on the rational cohomology of X. We know by Proposition 3.1 and Corollary 3.2 that intervals [ˆ0, p], p ∈ Π_{n}^{col,s}, are homotopic to a wedge of spheres of dimension rank(p) − 2. Let V_{p} be the subspace in L_{A_{n}^{col,s}} corresponding to p ∈ Π_{n}^{col,s}. Then codim(V_{p}) = 2 rank(p) – note that we consider real codimension. Therefore, [ˆ0, p] contributes to the cohomology of C^{n} \ U_{A_{n}^{col,s}} in dimension rank(p).

(i) There are exactly s + 1 orbits of S_{n} on elements of rank 1 in Π_{n}^{col,s} satisfying the conditions of assertion (ii) – one orbit of partitions with one block of size two and no blocks colored, s orbits of the partition 1| ⋯ |n with exactly one block colored.

(ii) Let p = (τ = B_{1}| ⋯ |B_{t}, (l_{1}, …, l_{t})) be some element of Π_{n}^{col,s}. Let G ≅ S_{|B_{1}|} × ⋯×S_{|B_{f−1}|} ×S_{e_{1}}[S_{1}]×⋯×S_{e_{n}}[S_{n}] be the stabilizer of p = (τ, (l_{1}, …, l_{t})) in S_{n}. By Corollary 4.3 the S_{n}-character on ind_{G}^{S_{n}} H̃_{n−t+f−2}([ˆ0, (τ, (l_{1}, …, l_{t}))]) is given by r_{|B_{1}|} ⋯ r_{|B_{f−1}|} · sgn_{e_{1}}[π_{1}] · 1_{e_{2}}[π_{2}] ⋯ for suitable parameters. Now, by the work of Sundaram [15, Corollary 2.3] the trivial S_{n}-representation appears in r_{|B_{1}|} ⋯ r_{|B_{f−1}|} · sgn_{e_{1}}[π_{1}] · 1_{e_{2}}[π_{2}] ⋯ if and only if e_{2} ≤ 1, e_{3} = ⋯ = e_{n} = 0 and in this case it appears exactly once. But this condition translates immediately to the condition stated in assertion (ii). It is clear that such partitions exist for all ranks.

Now, for s → ∞ the space M_{A_{n}^{col,s}} can be used to “approximate” the space of complex polynomials with no integral root. Note, that the combinatorics of A_{n}^{col,s} does not depend on the actual values of the excluded coordinates. Thus by choosing the sequence 0, −1, +1, −2, 2, … we exhaust the integers when s → ∞.

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