Séminaire Lotharingien de Combinatoire 77 (2018), Article B77e

DIVISIONALLY FREE RESTRICTIONS OF REFLECTION ARRANGEMENTS

GERHARD RÖHRLE

Abstract. We study some aspects of divisionally free arrangements which were recently introduced by Abe. Crucially, Terao’s conjecture on the combinatorial nature of freeness holds within this class. We show that while it is compatible with products, surprisingly, it is not closed under taking localizations. In addition, we determine all divisionally free restrictions of all reflection arrangements.

1. Introduction

The interplay between algebraic and combinatorial structures of hyperplane arrangements has been a driving force in the study of the field for many decades. At the very heart of these investigations lies Terao’s fundamental Conjecture 1.1 which asserts that the algebraic property of freeness of an arrangement is determined by purely combinatorial data.

Conjecture 1.1 ([OT92, Conj. 4.138]). For a fixed field, freeness of the arrangement 𝒜 only depends on its lattice L(𝒜), i.e. is combinatorial.

The conjecture is known to be sensitive to a change of the characteristic of the underlying field, cf. [Zie90, §4]. It is still open even for dimension 3.

Recently, T. Abe [Abe16] introduced a new class of free hyperplane arrangements, so called divisionally free arrangements DF (Definition 2.10). This properly encompasses the class of inductively free arrangements IF (Definition 2.5), cf. [Abe16, Thm. 1.6]. The relevance of this new notion is that Conjecture 1.1 is valid within DF, cf. [Abe16, Thm. 4.4(3)].

Each of the classes of free, inductively free and recursively free arrangements is compatible with the product construction for arrangements ([OT92, Prop. 4.28], [HR15, Prop. 2.10], [HRS17, Thm. 2]). We show that this also is the case for DF, cf. Proposition 2.12.

All the previously mentioned classes of free arrangements are known to be closed with respect to taking localizations ([OT92, Thm. 4.37], [HRS17, Thm. 1]). Unexpectedly, this fails for Abe’s new class DF, see Example 2.16.

Because of its relevance to Conjecture 1.1, it is important to know which arrangements from a given class belong to DF, e.g. see [Abe16, §6]. Reflection arrangements have had a pivotal role in the theory of hyperplane arrangements ever since. In [Abe16, Cor. 4.7], Abe determined all irreducible divisionally free reflection arrangements, see Theorem 3.1. We extend this classification to all restrictions of reflection arrangements in Theorem 3.3.

2010 Mathematics Subject Classification. 20F55, 52C35, 14N20.

Key words and phrases. hyperplane arrangements, Terao’s Conjecture, inductively free arrangements, reflection arrangements, restricted arrangements, divisionally free arrangements.

For general information about arrangements and reflection groups we refer the reader to [OS82] and [OT92]. In this article we use the classification and labeling of the irreducible unitary reflection groups due to Shephard and Todd, [ST54].

2. Preliminaries

2.1. Hyperplane Arrangements. Let 𝕂 be a field and let V = 𝕂^{ℓ}. By a hyperplane arrangement in V we mean a finite set 𝒜 of hyperplanes in V. Such an arrangement is denoted (𝒜 , V) or simply 𝒜. If dim V = ℓ we call 𝒜 an ℓ-arrangement. The number of elements in 𝒜 is given by |𝒜 |. The empty ℓ-arrangement is denoted by Φ_{ℓ}.

By L(𝒜) we denote the set of all nonempty intersections of elements of 𝒜, [OT92, Def. 1.12]. For X ∈ L(𝒜), we have two associated arrangements, firstly the subarrangement 𝒜_{X} := {H ∈ 𝒜 | X ⊆ H} ⊆ 𝒜 of 𝒜 and secondly, the restriction of 𝒜 to X, (𝒜^{X}, X), where 𝒜^{X} := {X ∩ H | H ∈ 𝒜 \ 𝒜_{X}}, [OT92, Def. 1.13]. Note that V belongs to L(𝒜) as the intersection of the empty collection of hyperplanes and 𝒜^{V} = 𝒜.

If 0 ∈ H for each H in 𝒜, then 𝒜 is called central. We only consider central arrangements.

Let H ∈ 𝒜 (for 𝒜 ≠ Φ_{ℓ}) and define 𝒜′ := 𝒜 \ {H}, and 𝒜′′ := 𝒜^{H}. Then (𝒜 , 𝒜′, 𝒜′′) is a triple of arrangements, [OT92, Def. 1.14].

The product 𝒜 = (𝒜_{1} × 𝒜_{2}, V_{1} ⊕ V_{2}) of two arrangements (𝒜_{1}, V_{1}), (𝒜_{2}, V_{2}) is defined by

𝒜 := 𝒜_{1} × 𝒜_{2} = {H_{1} ⊕ V_{2} | H_{1} ∈ 𝒜_{1}} ∪ {V_{1} ⊕ H_{2} | H_{2} ∈ 𝒜_{2}},

see [OT92, Def. 2.13]. In particular, |𝒜 | = |𝒜_{1}| + |𝒜_{2}|.

An arrangement 𝒜 is called reducible, if it is of the form 𝒜 = 𝒜_{1} × 𝒜_{2}, where 𝒜_{i} ≠ Φ_{0} for i = 1, 2, else 𝒜 is irreducible, [OT92, Def. 2.15].

If 𝒜 = 𝒜_{1} × 𝒜_{2} is a product, then by [OT92, Prop. 2.14] there is a lattice isomorphism

L(𝒜_{1}) × L(𝒜_{2}) ≅ L(𝒜) by (X_{1}, X_{2}) ↦ X_{1} ⊕ X_{2}.

It is easy to see that for X = X_{1} ⊕ X_{2} ∈ L(𝒜), we have

(2.1)  𝒜^{X} = 𝒜_{1}^{X_{1}} × 𝒜_{2}^{X_{2}}.

The characteristic polynomial χ(𝒜 , t) ∈ ℤ[t] of 𝒜 is defined by

χ(𝒜 , t) := ∑_{X∈L(𝒜)} μ(X)t^{dim X},

where μ is the Möbius function of L(𝒜), see [OT92, Def. 2.52].

If 𝒜 = 𝒜_{1} × 𝒜_{2} is a product, then, thanks to [OT92, Lem. 2.50],

(2.2)  χ(𝒜 , t) = χ(𝒜_{1}, t) · χ(𝒜_{2}, t).

2.2. Free Arrangements. Let S = S(V^{∗}) be the symmetric algebra of the dual space V^{∗} of V. If 𝒜 is an arrangement in V, then for every H ∈ 𝒜 we may fix α_{H} ∈ V^{∗} with H = ker(α_{H}). We call Q(𝒜) := ∏_{H∈𝒜} α_{H} ∈ S the defining polynomial of 𝒜.

The module of 𝒜 -derivations is the S-submodule of Der(S), the S-module of 𝕂-derivations of S, defined by

D(𝒜) := {θ ∈ Der(S) | θ(Q(𝒜)) ∈ Q(𝒜)S}.

The arrangement 𝒜 is said to be free if D(𝒜) is a free S-module.

If 𝒜 is a free ℓ-arrangement, then D(𝒜) admits an S-basis of ℓ homogeneous derivations θ_{1}, …, θ_{ℓ}, by [OT92, Prop. 4.18]. The exponents of the free arrangement 𝒜 are given by the multiset given by the polynomial degrees of the θ_{i}, exp 𝒜 := {pdeg θ_{1}, …, pdeg θ_{ℓ}}.

Terao’s basic Addition-Deletion Theorem plays a key role in the study of free arrangements.

Theorem 2.3 ([Ter80], [OT92, Thm. 4.51]). Suppose 𝒜 ≠ Φ_{ℓ} and let (𝒜 , 𝒜′, 𝒜′′) be a triple of arrangements. Then any two of the following statements imply the third:

(i) 𝒜 is free with exp 𝒜 = {b_{1}, …, b_{ℓ−1}, b_{ℓ}};

(ii) 𝒜′ is free with exp 𝒜′ = {b_{1}, …, b_{ℓ−1}, b_{ℓ} − 1};

(iii) 𝒜′′ is free with exp 𝒜′′ = {b_{1}, …, b_{ℓ−1}}.

The following is Terao’s celebrated Factorization Theorem for free arrangements.

Theorem 2.4 ([OT92, Thm. 4.137]). If 𝒜 is free with exp 𝒜 = {b_{1}, …, b_{ℓ−1}, b_{ℓ}}, then

χ(𝒜 , t) = ∏_{i=1}^{ℓ}(t − b_{i}).

2.3. Inductively Free Arrangements. An iterative application of the addition part of Theorem 2.3 leads to the class of inductively free arrangements.

Definition 2.5 ([OT92, Def. 4.53]). The class IF of inductively free arrangements is the smallest class of arrangements subject to

(i) Φ_{ℓ} ∈ IF for each ℓ ≥ 0;

(ii) if there exists a hyperplane H_{0} ∈ 𝒜 such that both 𝒜′ and 𝒜′′ belong to IF, and exp 𝒜′′ ⊆ exp 𝒜′, then 𝒜 also belongs to IF.

There is a hereditary version of IF, cf. [OT92, §6.4].

Definition 2.6. The arrangement 𝒜 is called hereditarily inductively free provided that 𝒜^{X} is inductively free for each X ∈ L(𝒜). Denote this class by HIF.

Note that if 𝒜 is hereditarily inductively free, it is inductively free as V ∈ L(𝒜) and 𝒜^{V} = 𝒜.

2.4. Reflection Arrangements. Let W ⊆ GL(V) be a finite, complex reflection group acting on the complex vector space V = ℂ^{ℓ}. The reflection arrangement of W in V is the hyperplane arrangement 𝒜 (W) consisting of the reflecting hyperplanes of the elements in W acting as reflections on V.

Terao [Ter80] has shown that every reflection arrangement 𝒜 (W) is free and that the exponents of 𝒜 (W) coincide with the coexponents of W, cf. [OT92, Prop. 6.59 and Thm. 6.60].

We recall the classification of the inductively free reflection arrangements from [HR15].

Theorem 2.7 ([HR15, Thms. 1.1 and 1.2]). Let W be a finite, irreducible, complex reflection group with reflection arrangement 𝒜 (W). Then the following hold:

(i) 𝒜 (W) is inductively free if and only if W does not admit an irreducible factor isomorphic to a monomial group G(r, r, ℓ) for r, ℓ ≥ 3, G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, or G_{34}.

(ii) 𝒜 (W) is inductively free if and only if it is hereditarily inductively free.

2.5. Restrictions of Reflection Arrangements. Orlik and Solomon defined intermediate arrangements 𝒜_{ℓ}^{k}(r) in [OS82, §2] (cf. [OT92, §6.4]) which interpolate between the reflection arrangements of G(r, r, ℓ) and G(r, 1, ℓ). They show up as restrictions of the reflection arrangement of G(r, r, ℓ), [OS82, Prop. 2.14] (cf. [OT92, Prop. 6.84]).

For ℓ ≥ 2 and 0 ≤ k ≤ ℓ the defining polynomial of 𝒜_{ℓ}^{k}(r) is given by

Q(𝒜_{ℓ}^{k}(r)) = x_{1} ⋯ x_{k} ∏_{\begin{matrix}1≤i<j≤ℓ \\ 0≤n<r \end{matrix}}(x_{i} − ζ^{n}x_{j}),

where ζ is a primitive r^{th} root of unity, so that 𝒜_{ℓ}^{ℓ}(r) = 𝒜 (G(r, 1, ℓ)) and 𝒜_{ℓ}^{0}(r) = 𝒜 (G(r, r, ℓ)). For k ≠ 0, ℓ, these are not reflection arrangements themselves.

We recall the classification of the inductively free restrictions of all reflection arrangements.

Theorem 2.8 ([AHR14, Thms. 1.2 and 1.3]). Let W be a finite, irreducible, complex reflection group with reflection arrangement 𝒜 (W) and let V ≠ X ∈ L(𝒜 (W)).

(a) 𝒜 (W)^{X} is inductively free if and only if one of the following holds:

(i) 𝒜 is inductively free;

(ii) W = G(r, r, ℓ), r ≥ 3 and 𝒜 (W)^{X} ≅ 𝒜_{p}^{k}(r), for p = dim X and p − 2 ≤ k ≤ p;

(iii) W = G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, or G_{34} and X ∈ L(𝒜 (W)) with dim X ≤ 3.

(b) 𝒜 (W)^{X} is inductively free if and only if it is hereditarily inductively free.

2.6. Divisionally Free Arrangements. First we recall the key result from [Abe16].

Theorem 2.9 ([Abe16, Thm. 1.1]). Let 𝒜 ≠ Φ_{ℓ}. Suppose there is a hyperplane H in 𝒜 such that the restriction 𝒜^{H} is free and that χ(𝒜^{H}, t) divides χ(𝒜 , t). Then 𝒜 is free.

Theorem 2.9 can be viewed as a strengthening of the addition part of Theorem 2.3. An iterative application leads to the class DF.

Definition 2.10 ([Abe16, Def. 1.5]). An ℓ-arrangement 𝒜 is called divisionally free if ℓ ≤ 2, 𝒜 = Φ_{ℓ}, or else there is a sequence of consecutive restrictions of arrangements starting with 𝒜 and ending in a 2-arrangement such that the successive characteristic polynomials divide one another. That is, there is a sequence of arrangements 𝒜 = 𝒜_{ℓ}, 𝒜_{ℓ−1}, …, 𝒜_{2} such that for each i = 3, …, ℓ there is an H_{i} in 𝒜_{i} so that 𝒜_{i}^{H_{i}} = 𝒜_{i−1} and χ(𝒜_{i}^{H_{i}}, t) divides χ(𝒜_{i}, t). Denote this class by DF.

Thanks to Theorem 2.9 and the fact that any 2-arrangement is free, any 𝒜 in DF is free.

In [Abe16, Thm. 1.6], Abe observed that IF ⊊ DF. The reflection arrangement of the complex reflection group G_{31} is divisionally free but not inductively free.

Theorem 2.11 ([Abe16, Thm. 5.6]). Let r, ℓ ≥ 3. Then 𝒜_{ℓ}^{k}(r) ∈ DF if and only if k ≠ 0.

It is immediate from Theorem 2.11 and [OS82, Prop. 2.11] (cf. [OT92, Prop. 6.82]) that the class DF is not closed under restrictions.

Each of the classes of free, inductively free and recursively free arrangements is compatible with the product construction for arrangements, cf. [OT92, Prop. 4.28], [HR15, Prop. 2.10], [HRS17, Thm. 2]. We observe that this also holds for the class DF.

Proposition 2.12. Let (𝒜_{1}, V_{1}), (𝒜_{2}, V_{2}) be two arrangements. Then 𝒜 = (𝒜_{1}×𝒜_{2}, V_{1}⊕V_{2}) is divisionally free if and only if both 𝒜_{1} and 𝒜_{2} are divisionally free.

Proof. First suppose that both 𝒜_{1} and 𝒜_{2} are divisionally free. Then we claim that also 𝒜 is so. We argue via induction on |𝒜 |. If both 𝒜_{1} and 𝒜_{2} are empty, there is nothing to show. So suppose that |𝒜 | ≥ 1 and that the claim holds for any product of divisionally free arrangements with fewer than |𝒜 | hyperplanes. Without loss of generality there exists an H_{1} in 𝒜_{1} such that 𝒜_{1}^{H_{1}} belongs to DF and χ(𝒜_{1}^{H_{1}}, t) divides χ(𝒜_{1}, t). Letting H = H_{1} ⊕ V_{2} ∈ 𝒜, by (2.1), 𝒜^{H} = 𝒜_{1}^{H_{1}} × 𝒜_{2} is a product of divisionally free arrangements with |𝒜^{H}| < |𝒜 |. So, by our induction hypothesis, 𝒜^{H} is divisionally free. In addition, since χ(𝒜^{H}, t) = χ(𝒜_{1}^{H_{1}}, t) · χ(𝒜_{2}, t) divides χ(𝒜_{1}, t) · χ(𝒜_{2}, t) = χ(𝒜 , t), cf. (2.2), we infer that 𝒜 belongs to DF.

Conversely, suppose that 𝒜 = 𝒜_{1} × 𝒜_{2} belongs to DF. We claim that then both 𝒜_{1} and 𝒜_{2} also belong to DF. Again we argue by induction on |𝒜 |. If both 𝒜_{1} and 𝒜_{2} are empty, there is nothing to show. So suppose that |𝒜 | ≥ 1 and that the claim holds for any product in DF with fewer than |𝒜 | hyperplanes. Without loss of generality we may assume that there is an H = H_{1} ⊕ V_{2} in 𝒜 such that 𝒜^{H} belongs to DF and χ(𝒜^{H}, t) divides χ(𝒜 , t). Since |𝒜^{H}| < |𝒜 | and 𝒜^{H} = 𝒜_{1}^{H_{1}} × 𝒜_{2}, both 𝒜_{1}^{H_{1}} and 𝒜_{2} belong to DF, by our induction hypothesis. Moreover, since χ(𝒜^{H}, t) = χ(𝒜_{1}^{H_{1}}, t)·χ(𝒜_{2}, t) and χ(𝒜 , t) = χ(𝒜_{1}, t)·χ(𝒜_{2}, t), cf. (2.2), it follows that χ(𝒜_{1}^{H_{1}}, t) divides χ(𝒜_{1}, t). Therefore, also 𝒜_{1} belongs to DF. □

There is a hereditary version of DF.

Definition 2.13 ([Abe16, Def. 5.7(2)]). The arrangement 𝒜 is called hereditarily divisionally free provided that 𝒜^{X} is divisionally free for each X ∈ L(𝒜). We denote this class by HDF.

Remark 2.14. (i). Clearly, since IF ⊆ DF, we have HIF ⊆ HDF.

(ii). Note that HDF ⊊ DF. For, in [HR15, Ex. 2.16], we constructed an inductively free arrangement 𝒜 which admits a hyperplane H such that 𝒜^{H} is not free. In particular, 𝒜 ∈ DF \ HDF.

The compatibility with products from Proposition 2.12 descends to HDF. The proof follows readily from Proposition 2.12 and (2.1).

Corollary 2.15. Let 𝒜_{1}, 𝒜_{2} be two arrangements. Then 𝒜 = 𝒜_{1} × 𝒜_{2} is hereditarily divisionally free if and only if both 𝒜_{1} and 𝒜_{2} are hereditarily divisionally free.

While the classes of free, inductively free and recursively free arrangements are closed under taking localizations, cf. [OT92, Thm. 4.37], [HRS17, Thm. 1], unexpectedly, this property fails for DF, as the following example illustrates.

Example 2.16. Let 𝒜 = 𝒜_{ℓ}^{1}(r) for r ≥ 3 and ℓ ≥ 4. Let

X = ⋂_{\begin{matrix}2≤i<j≤ℓ \\ 0≤n<r \end{matrix}} ker(x_{i} − ζ^{n}x_{j}),

where ζ is a primitive r^{th} root of unity. Then 𝒜_{X} ≅ 𝒜_{ℓ−1}^{0}(r). By Theorem 2.11, 𝒜 is divisionally free but 𝒜_{X} is not.

Remark 2.17. If 𝒜 ≠ Φ_{ℓ} is inductively free and (𝒜 , 𝒜′, 𝒜′′) is a triple as in Definition 2.5(ii), then each member of (𝒜 , 𝒜′, 𝒜′′) belongs to IF. In contrast, if 𝒜 and 𝒜′′ are consecutive members in a sequence as in Definition 2.10, then the deletion 𝒜′ in the corresponding triple (𝒜 , 𝒜′, 𝒜′′) need not belong to DF. For instance, let 𝒜 = 𝒜_{ℓ}^{1}(r) for r, ℓ ≥ 3 and H = ker(x_{1}). Then 𝒜^{H} ≅ 𝒜_{ℓ−1}^{ℓ−1}(r), by [OS82, Prop. 2.11]. It follows from [OS82, Prop. 2.13] (or else from [OT92, Prop. 6.85] and Theorem 2.4) that χ(𝒜^{H}, t) divides χ(𝒜 , t). Consequently, thanks to Theorem 2.11, 𝒜 and 𝒜′′ are successive terms in a sequence as in Definition 2.10. However, 𝒜′ = 𝒜_{ℓ}^{0}(r) is not divisionally free, by Theorem 2.11.

3. Divisionally Free Reflection Arrangements

In view of Theorem 2.7(i) and Proposition 2.12, we can restate Abe’s classification of the divisionally free reflection arrangements [Abe16, Cor. 4.7] as follows.

Theorem 3.1. For W a finite, complex reflection group, its reflection arrangement 𝒜 (W) is divisionally free if and only if W does not admit an irreducible factor isomorphic to a monomial group G(r, r, ℓ) for r, ℓ ≥ 3, G_{24}, G_{27}, G_{29}, G_{33}, or G_{34}.

The counterpart to Theorem 2.7(ii) also holds in the setting of divisional freeness:

Theorem 3.2. The reflection arrangement of a finite, complex reflection group is divisionally free if and only if it is hereditarily divisionally free.

Proof. The forward implication follows from Proposition 2.12, Theorems 2.7(ii) and 3.1, and [Abe16, Prop. 5.8]. The reverse implication is obvious. □

We now consider restrictions of reflection arrangements. Thanks to Proposition 2.12 and (2.1), the question of divisional freeness of 𝒜^{X} reduces readily to the case when 𝒜 is irreducible, so we may assume that W is irreducible. In view of Theorem 2.8(a), we can formulate our classification as follows:

Theorem 3.3. Let W be a finite, irreducible, complex reflection group with reflection arrangement 𝒜 (W) and let V ≠ X ∈ L(𝒜 (W)). Then 𝒜 (W)^{X} is divisionally free if and only if one of the following holds:

(i) 𝒜 (W)^{X} is inductively free;

(ii) W = G(r, r, ℓ) and 𝒜 (W)^{X} ≅ 𝒜_{p}^{k}(r), where p = dim X and 1 ≤ k ≤ p − 3;

(iii) W = G_{33} or G_{34} and dim X = 4.

Proof. If 𝒜 (W)^{X} is inductively free, the result follows, since IF ⊆ DF.

Now suppose that 𝒜 (W)^{X} is not inductively free. For W = G(r, r, ℓ), every restriction is of the form 𝒜 (W)^{X} ≅ 𝒜_{p}^{k}(r), where p = dim X and 1 ≤ k ≤ p, by [OS82, Prop. 2.14]. Thus (ii) follows from Theorems 2.8(a)(ii) and 2.11.

For W = G_{33} or G_{34} the result follows from [OS82, Thm. 1.2], [OS82, Tables 10, 11], along with Theorem 2.8(a)(iii). □

In view of Proposition 2.12 and Theorem 2.8, we can restate Theorem 3.3 as follows.

Theorem 3.4. Let W be a finite, irreducible, complex reflection group and let V ≠ X ∈ L(𝒜 (W)). Then 𝒜 (W)^{X} is divisionally free unless W = G_{34} and dim X = 5.

Finally, we show that Theorem 3.2 extends to restrictions.

Theorem 3.5. Let W be a finite, complex reflection group with reflection arrangement 𝒜 (W) and let V ≠ X ∈ L(𝒜 (W)). The restricted arrangement 𝒜 (W)^{X} is divisionally free if and only if it is hereditarily divisionally free.

Proof. The forward implication follows from Corollary 2.15, [OS82, Prop. 2.11, Prop. 2.14], and Theorems 3.2, 3.3, and 2.8(b). The reverse implication is obvious. □

Acknowledgments: We acknowledge support from the DFG-priority program SPP1489 “Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory”.

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Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany

E-mail address: gerhard.roehrle@rub.de