2. Preliminaries

2.1. Hyperplane arrangements

Let $\mathbb K$ be a field and let $V = {\mathbb K}^{\ell}$ be an ${\ell}$ -dimensional $\mathbb K$ -vector space. A hyperplane arrangement is a pair $({\mathscr A}, V)$ , where $\mathscr A{\kern1.5pt}$ is a finite collection of hyperplanes in $V$ . Usually, we simply write $\mathscr A{\kern1.5pt}$ in place of $({\mathscr A}, V)$ . By ${\Phi_\ell}$ we denote the empty arrangement in $V$ .

The lattice $L({\mathscr A}{\kern1.5pt})$ of $\mathscr A{\kern1.5pt}$ is the set of subspaces of $V$ of the form $H_1\cap \ldots \cap H_i$ where $\{ H_1, \ldots, H_i\}$ is a subset of $\mathscr A$ . For $X \in L({\mathscr A}{\kern1.5pt})$ , we have two associated arrangements, firstly ${\mathscr A}_X :\!=\{H \in {\mathscr A} \mid X \subseteq H\} \subseteq {\mathscr A}$ , the localization of $\mathscr A{\kern1.5pt}$ at $X$ , and secondly, the restriction of $\mathscr A{\kern1.5pt}$ to $X$ , $({\mathscr A}^{{\kern1.5pt}X},X)$ , where ${\mathscr A}^{{\kern1.5pt}X} :\!= \{ X \cap H \mid H \in {\mathscr A} \setminus {\mathscr A}_X\}$ . The lattice $L({\mathscr A}{\kern1.5pt})$ is a partially ordered set by reverse inclusion: $X \le Y$ provided $Y \subseteq X$ for $X,Y \in L({\mathscr A}{\kern1.5pt})$ .

A property for arrangements is said to be local if it is passed to every localization of the ambient arrangement. For instance, freeness, asphericity, and factoredness are all local (cf. [Reference Orlik and Terao34, Thm. 4.37], [Reference Paris35, Lem. 1.1], and the proof of [Reference Terao46, Cor. 2.11], respectively). In contrast, this fails for formal arrangements of rank at least 4 (cf. the example following [Reference Yuzvinsky47, Def. 2.3]).

Throughout, we only consider arrangements $\mathscr A{\kern1.5pt}$ such that $0 \in H$ for each $H$ in $\mathscr A$ . These are called central. In that case the center $T({\mathscr A}{\kern1.5pt}) :\!= \cap _{H \in {\mathscr A}} H$ of $\mathscr A{\kern1.5pt}$ is the unique maximal element in $L({\mathscr A}{\kern1.5pt})$ with respect to the partial order. A rank function on $L({\mathscr A}{\kern1.5pt})$ is given by $r(X) :\!= \operatorname {codim}_V(X)$ . The rank of $\mathscr A{\kern1.5pt}$ is defined as $r({\mathscr A}{\kern1.5pt}) :\!= r(T({\mathscr A}{\kern1.5pt}))$ .

The Poincaré polynomial $\pi ({\mathscr A},t) \in {\mathbb Z}[t]$ of $\mathscr A{\kern1.5pt}$ is defined by

\begin{equation*} \pi ({\mathscr A},t)\, :\!= \sum _{X \in L({\mathscr A}{\kern1.5pt})} \mu (X)({-}t)^{r(X)}, \end{equation*}

and the characteristic polynomial $\chi ({\mathscr A},t) \in {\mathbb Z}[t]$ of $\mathscr A{\kern1.5pt}$ is defined by

\begin{equation*} \chi ({\mathscr A},t)\, :\!= t^\ell \pi ({\mathscr A},-t^{-1}) = \sum _{X \in L({\mathscr A}{\kern1.5pt})} \mu (X)t^{\dim X}, \end{equation*}

where $\mu$ is the Möbius function of $L({\mathscr A}{\kern1.5pt})$ , see [Reference Orlik and Terao34, Def. 2.48, Def. 2.52].

We recall the concept of a generic arrangement from [Reference Orlik and Terao34, Def. 5.22].

Definition 2.1. An $\ell$ -arrangement $\mathscr A{\kern1.5pt}$ with ${r}({\mathscr A}{\kern1.5pt})=r$ is called generic if every subarrangement $\mathscr B$ of $\mathscr A{\kern1.5pt}$ of cardinality $\ell$ is linearly independent and ${|\mathscr A{\kern1.5pt}| \gt r}$ , [Reference Orlik and Terao34, Def. 5.22].

2.2. Supersolvable arrangements

Let $\mathscr A{\kern1.5pt}$ be an arrangement. Following [Reference Orlik and Terao34, §2], we say that $X \in L({\mathscr A}{\kern1.5pt})$ is modular provided $X + Y \in L({\mathscr A}{\kern1.5pt})$ for every $Y \in L({\mathscr A}{\kern1.5pt})$ , cf. [Reference Orlik and Terao34, Def. 2.32, Cor. 2.26]. The following notion is due to Stanley [Reference Stanley42].

Definition 2.2. Let $\mathscr A{\kern1.5pt}$ be a central (and essential) $\ell$ -arrangement. We say that $\mathscr A{\kern1.5pt}$ is supersolvable provided there is a maximal chain

\begin{equation*} V = X_0 \lt X_1 \lt \ldots \lt X_{\ell -1} \lt X_\ell = \{0\} \end{equation*}

of modular elements $X_i$ in $L({\mathscr A}{\kern1.5pt})$ .

Remark 2.3.

1. (i) By [Reference Orlik and Terao34 , Ex. 2.28], $V$ , $\{0\}$ and the members in $\mathscr A{\kern1.5pt}$ are always modular in $L({\mathscr A}{\kern1.5pt})$ . It follows that all $0$ -,  $1$ -, and $2$ -arrangements are supersolvable.

2. (ii) Supersolvability is a local property [Reference Stanley42 , Prop. 3.2].

2.3. Free arrangements

Free arrangements play a crucial role in the theory of arrangements; see [Reference Orlik and Terao34, §4] for the definition and basic properties. If $\mathscr A{\kern1.5pt}$ is free, then we can associate with $\mathscr A{\kern1.5pt}$ the multiset of its exponents, denoted $\exp {\mathscr A}$ .

Remark 2.4.

1. (i) Generic arrangements are not free, for example see [Reference Rose and Terao39, §4.4].

2. (ii) Freeness is a local property [Reference Orlik and Terao34, Thm. 4.37].

Terao’s Factorization Theorem [Reference Terao44] shows that the Poincaré polynomial of a free arrangement $\mathscr A{\kern1.5pt}$ factors into linear terms given by the exponents of $\mathscr A{\kern1.5pt}$ (cf. [Reference Orlik and Terao34, Thm. 4.137]):

Theorem 2.5. Suppose that $\mathscr A{\kern1.5pt}$ is free with $\exp {\mathscr A} = \{ b_1, \ldots, b_\ell \}$ . Then

\begin{equation*} \pi ({\mathscr A},t) = \prod _{i=1}^\ell (1 + b_i t). \end{equation*}

Terao’s celebrated Addition-Deletion Theorem [Reference Terao43] plays a fundamental role in the study of free arrangements, [Reference Orlik and Terao34, Thm. 4.51].

Theorem 2.6. Suppose that ${\mathscr A} \ne \Phi _\ell$ . Let ${H_0 \in \mathscr A}$ . Let ${(\mathscr A, \mathscr A{\kern1.5pt}' = \mathscr A \setminus \{H_0\}, \mathscr A{\kern1.5pt}'' = \mathscr A^{{\kern1.5pt}H_0})}$ be a triple of arrangements. Then any two of the following statements imply the third:

1. (i) $\mathscr A{\kern1.5pt}$ is free with $\exp {\mathscr A} = \{ b_1, \ldots, b_{\ell -1}, b_\ell \}$ ;

2. (ii) ${\mathscr A}{\kern1.5pt}'$ is free with $\exp {\mathscr A}{\kern1.5pt}' = \{ b_1, \ldots, b_{\ell -1}, b_\ell -1\}$ ;

3. (iii) ${\mathscr A}{\kern1.5pt}''$ is free with $\exp {\mathscr A}{\kern1.5pt}'' = \{ b_1, \ldots, b_{\ell -1}\}$ .

Theorem 2.6 motivates the notion of an inductively free arrangement, [Reference Orlik and Terao34, Def. 4.53].

Definition 2.7. The class $\mathcal{IF}$ of inductively free arrangements is the smallest class of arrangements subject to

1. (i) $\Phi _\ell \in {\mathcal{IF}}$ for each $\ell \ge 0$ ;

2. (ii) if there exists a hyperplane $H_0 \in {\mathscr A}$ such that both ${\mathscr A}{\kern1.5pt}'$ and ${\mathscr A}{\kern1.5pt}''$ belong to $\mathcal{IF}$ , and $\exp {\mathscr A}{\kern1.5pt} '' \subseteq \exp {\mathscr A}{\kern1.5pt}'$ , then $\mathscr A{\kern1.5pt}$ also belongs to $\mathcal{IF}$ .

Remark 2.8.

1. (i) Inductively free arrangements are free. However, the latter class properly contains the former, cf. [Reference Orlik and Terao34 , Ex. 4.59].

2. (ii) Inductive freeness is also a local property, thanks to [Reference Hoge, Röhrle and Schauenburg26 , Thm 1.1].

3. (iii) Supersolvable arrangements are inductively free, [Reference Orlik and Terao34 , Thm. 4.58].

2.6. Simplicial arrangements

A real arrangement $\mathscr A{\kern1.5pt}$ is simplicial provided each chamber of the complement of $\mathscr A{\kern1.5pt}$ is an open simplicial cone in the ambient space. Simpliciality is a combinatorial property. For, thanks to [Reference Cuntz and Geis10, Cor. 2.4], a central essential real $\ell$ -arrangement $\mathscr A{\kern1.5pt}$ is simplicial if and only if

(2.1) \begin{equation} \ell \cdot \chi ({\mathscr A}, -1) + 2\sum _{H \in {\mathscr A}} \chi ({\mathscr A}^{{\kern2pt}H}, -1) = 0. \end{equation}

Remark 2.19. Simpliciality is preserved under localizations, see [Reference Cuntz and Mücksch13 , Lem. 2.17(1)].

3. Proof of Theorem 1.2

We begin by identifying the small rank hyperpolygonal arrangements ${\mathscr H}_n$ with known ones. Clearly, ${\mathscr H}_2$ is just the reflection arrangement of the Weyl group of type $B_2$ .

It follows from the next lemma that ${\mathscr H}_3$ is linearly isomorphic to the connected subgraph arrangement ${\mathscr A}_G$ , where $G = C_3$ is the cycle graph on three vertices. See [Reference Cuntz and Kühne12] for the class of connected subgraph arrangements.

Proof. One checks that (up to scalar multiples) the map on linear forms

\begin{equation*}\varphi\ :\ {\mathbb R}^3 \rightarrow {\mathbb R}^3 \ \ \ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \mapsto \begin{bmatrix} x_1+x_2 \\ x_1+x_3 \\ x_2+x_3 \end{bmatrix}\end{equation*}

gives a linear isomorphism between ${\mathscr H}_3$ and ${\mathscr A}_{C_3}$ .

We also note that ${\mathscr H}_3$ is linearly isomorphic to the cone over the Shi arrangement of type $A_2$ , by means of Lemma 3.1 and [Reference Cuntz and Kühne12, Prop. 3.1].

Our next lemma shows that ${\mathscr H}_4$ is linearly isomorphic to the reflection arrangement ${\mathscr A}(D_4)$ of the Weyl group of type $D_4$ .

Proof. One checks that (up to scalar multiples) the map on linear forms

\begin{equation*}\varphi\ :\ {\mathbb R}^4 \rightarrow {\mathbb R}^4 \ \ \ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \mapsto \begin{bmatrix} x_1+x_2+x_3 \\ x_2 \\ x_2+x_3+x_4 \\ -x_1-x_2-x_4 \end{bmatrix}\end{equation*}

gives a linear isomorphism between ${\mathscr H}_4$ and ${\mathscr A}(D_4)$ .

Our next lemma gives that the reverse implications in the statements in Theorem 1.2 hold for $n\ge 6$ .

Thus, it remains to show the forward implications in the statements in Theorem 1.2 and the remaining reverse implications for $n \le 5$ .

Part (i). Thanks to Remark 2.3 (i), ${\mathscr H}_2$ is supersolvable. It follows from Lemma 3.1 and [Reference Cuntz and Kühne12, Cor. 8.11] that ${\mathscr H}_3$ is not supersolvable. By [Reference Hoge and Röhrle25, Lem. 3.2], ${\mathscr A}(D_4)$ is not factored (thus not inductively factored), thus, thanks to Lemma 3.2, neither is ${\mathscr H}_4$ , so the latter is not supersolvable, by Remark 2.18 (iii). Thanks to [Reference Edelman and Reiner17], ${\mathscr H}_5$ is not inductively free, thus by Remark 2.8 (iii) it is not supersolvable. Thus Theorem 1.2 (i) now follows from Lemma 3.3.

Part (ii). It follows from [Reference Giordani, Möller, Mücksch and Röhrle23, Thm. 1.7] that ${\mathscr A}_{C_3}$ is inductively factored, thus so is ${\mathscr H}_3$ , by Lemma 3.1. We have already observed above that ${\mathscr H}_4$ is not inductively factored. Since ${\mathscr H}_5$ is not inductively free, it is also not inductively factored, by Remark 2.18 (i). Consequently, Theorem 1.2 (ii) follows again from Lemma 3.3.

Part (iii). For $n \le 4$ , the arrangements ${\mathscr H}_n$ are inductively free: For $n \le 3$ , this follows from Theorem 1.2 (ii) and Remark 2.18 (i). For ${\mathscr H}_4$ this follows from Lemma 3.2 and [Reference Jambu and Terao28, Ex. 2.6]. Edelman and Reiner [Reference Edelman and Reiner17] have observed that ${\mathscr H}_5$ is free but not inductively free. Consequently, Theorem 1.2 (iii) follows from Lemma 3.3.

Part (iv). By Theorem 1.2 (iii) and [Reference Edelman and Reiner17], ${\mathscr H}_n$ is free for $n \le 5$ . Whence Theorem 1.2 (iv) is once again a consequence of Lemma 3.3.

Part (v). With the aid of the formula (2.1), it is straightforward to check that ${\mathscr H}_n$ is simplicial for $n \le 4$ and that ${\mathscr H}_5$ is no longer simplicial. So Theorem 1.2 (v) follows from Lemma 3.3.

Finally, Theorem 1.2 (vi) is immediate from Lemma 3.3.

In closing this section, we show that ${\mathscr H}_5$ is still recursively free, a concept originally due to Ziegler [Reference Ziegler48]; see [Reference Orlik and Terao34, Def. 4.60].

4. Projective uniqueness: proof of Theorem 1.5

We recall some results from [Reference Giordani, Möller, Mücksch and Röhrle23]. Let $\mathscr A{\kern1.5pt}$ be an $\mathbb R$ -arrangement. We specify what we mean by a subarrangement of $\mathscr A{\kern1.5pt}$ being generated by a subarrangement of $\mathscr A$ .

Definition 4.2. Let $\varnothing \ne S \subseteq {\mathscr A}$ . Set $\operatorname {Gen}_0({\mathscr A},S){\kern2pt} :\!= S$ and inductively

\begin{equation*}\operatorname {Gen}_{i+1}({\mathscr A},S){\kern2pt} :\!= \left \{H \in {\mathscr A} \mid \exists \ J \subseteq L(\operatorname {Gen}_i({\mathscr A},S)) \,:\, H = \sum _{X \in J} X \right \}\end{equation*}

for $i \ge 0$ . Then we say that

\begin{equation*}\langle S \rangle _{\mathscr A}{\kern2pt} :\!= \bigcup _{i\ge 0} \operatorname {Gen}_i({\mathscr A},S) \subseteq {\mathscr A}\end{equation*}

is the subarrangement of $\mathscr A{\kern1.5pt}$ generated by $S$ . If $\langle S \rangle _{\mathscr A} = {\mathscr A}$ , then we say that $S$ generates $\mathscr A$ .

The following is a consequence of Lemma 4.3.

Finally, the following result gives Theorem 1.5.

Proposition 4.5. Let $ n \ge 1$ and let

\begin{equation*}S{\kern2pt} :\!= \{\ker x_1, \ldots, \ker x_{n-1}, \ker (x_1 + \ldots + x_n), \ker (x_1 + \ldots + x_{n-1} - x_n)\} \subseteq {\mathscr H}_n.\end{equation*}

Then $\langle S \rangle _{{\mathscr H}_n} = {\mathscr H}_n$ . In particular, ${\mathscr H}_n$ is projectively unique over $\mathbb R$ .

Proof. Let $\varnothing \ne I \subseteq [n]$ . Without loss, we may assume that $n \notin I$ . For $i \in [n-1]$ , set $H_i :\!= \ker x_i$ , and for $\alpha :\!= x_1 + \ldots + x_n$ and $\beta :\!= x_1 + \ldots + x_{n-1} - x_n$ , set $H_\alpha :\!= \ker \alpha$ and $H_\beta :\!= \ker \beta$ . Then $H_I = \ker \left (\sum _{i \in I}x_i - \sum _{j \in [n]\setminus I}x_j\right ) = \ker \gamma$ , where

(4.6) \begin{equation} \gamma = - \alpha + 2 \sum _{i \in I}x_i = \beta - 2 \sum _{j \in [n-1]\setminus I}x_j. \end{equation}

It follows from (4.6) that

\begin{align*} X & {\kern2pt}:\!= H_\alpha \cap \bigcap _{i \in I} H_i \subseteq H_I \text{ and } \\ Y &{\kern2pt} :\!= H_\beta \cap \bigcap _{j \in [n-1]\setminus I} H_j \subseteq H_I. \end{align*}

Since $\dim X = n - |I| - 1$ , $\dim Y = |I|$ , and $\dim (X \cap Y) = 0$ , we have $H_I = X + Y$ . Since both $X$ and $Y$ belong to the lattice of intersections of the subarrangement $S$ of ${\mathscr H}_n$ , we infer $H_I = X + Y \in \langle S \rangle _{{\mathscr H}_n}$ . Consequently, the result follows thanks to Proposition 4.4.

5. Combinatorial Formality: proof of Theorem 1.6

A property for arrangements is said to be combinatorial if it only depends on the intersection lattice of the underlying arrangement. Yuzvinsky [Reference Yuzvinsky47, Ex. 2.2] demonstrated that formality is not combinatorial, answering a question raised by Falk and Randell [Reference Falk and Randell21] in the negative. Yuzvinsky’s insight motivates the following notion from [Reference Möller, Mücksch and Röhrle30].

The following definitions, which are originally due to Falk for matroids [Reference Falk19], were adapted for arrangements in [Reference Möller, Mücksch and Röhrle30, §2.4]. Let ${\mathscr B} \subset {\mathscr A}$ be a subset of hyperplanes. We say $\mathscr B$ is closed if ${\mathscr B} ={\mathscr A}_Y$ for $Y=\bigcap \limits _{H\in {\mathscr B}} H$ . We call ${\mathscr B}$ line-closed if for every pair $H,H'\in {\mathscr B}$ of hyperplanes, we have ${\mathscr A}_{H\cap H'}\subset {\mathscr B}$ . The line-closure ${\operatorname {lc}}({\mathscr B})$ of ${\mathscr B}$ is defined as the intersection of all line-closed subsets of $\mathscr A{\kern1.5pt}$ containing $\mathscr B$ . The arrangement $\mathscr A{\kern1.5pt}$ is called line-closed if every line-closed subset of $\mathscr A{\kern1.5pt}$ is closed. With these notions, we have the following criterion for combinatorial formality, see [Reference Falk19, Cor. 3.8], [Reference Möller, Mücksch and Röhrle30, Prop. 3.2]:

A subset ${\mathscr B} \subseteq {\mathscr A}$ as in Proposition 5.2 is called an lc-basis of $\mathscr A$ .

There is a stronger notion of formality for an arrangement $\mathscr A$ , that of $k$ -formality for $1 \le k \le \operatorname {rk}({\mathscr A}{\kern1.5pt})$ due to Brandt and Terao [Reference Brandt and Terao9]. In view of Theorem 1.6 and in view of the fact that all free arrangements are not just formal but are $k$ -formal for all $k$ , by [Reference Brandt and Terao9, Thm. 4.15], one might ask for this stronger notion of $k$ -formality among hyperpolygonal arrangements. Computational evidence for further non-free hyperpolygonal arrangements suggests the following.

6. Rank-generating functions of the poset of regions

Let $\mathscr A{\kern1.5pt}$ be a hyperplane arrangement in the real vector space $V={\mathbb R}^\ell$ . A region of $\mathscr A{\kern1.5pt}$ is a connected component of the complement $M({\mathscr A}{\kern2pt}) :\!= V \setminus \cup _{H \in {\mathscr A}}H$ of $\mathscr A$ . Let ${\mathscr R}{\kern2pt} :\!= {\mathscr R}({\mathscr A}{\kern1.5pt})$ be the set of regions of $\mathscr A$ . For $R, R' \in {\mathscr R}$ , we let ${\mathcal S}(R,R')$ denote the set of hyperplanes in $\mathscr A{\kern1.5pt}$ separating $R$ and $R'$ . Then with respect to a choice of a fixed base region $B$ in $\mathscr R$ , we can partially order $\mathscr R$ as follows:

\begin{equation*} R \le R' \quad \text{ if } \quad {\mathcal S}(B,R) \subseteq {\mathcal S}(B,R'). \end{equation*}

Endowed with this partial order, we call $\mathscr R$ the poset of regions of $\mathscr A{\kern1.5pt}$ (with respect to $B$ ) and denote it by $P({\mathscr A}, B)$ . This is a ranked poset of finite rank, where $\operatorname {rk}(R) :\!= |{\mathcal S}(B,R)|$ , for $R$ a region of $\mathscr A$ , [Reference Edelman16, Prop. 1.1]. The rank-generating function of $P({\mathscr A}, B)$ is defined to be the following polynomial in ${\mathbb Z}_{\ge 0}[t]$

\begin{equation*} \zeta (P({\mathscr A},B);\, t){\kern2pt} :\!= \sum _{R \in {\mathscr R}}t^{\operatorname {rk}(R)}. \end{equation*}

This poset along with its rank-generating function was introduced by Edelman [Reference Edelman16].

Thanks to work of Björner, Edelman, and Ziegler [Reference Björner, Edelman and Ziegler8, Thm. 4.4] (see also Paris [Reference Paris37]), respectively Jambu and Paris [Reference Jambu and Paris27, Prop. 3.4, Thm. 6.1], in case of a real arrangement $\mathscr A{\kern1.5pt}$ which is supersolvable, respectively inductively factored, there always exists a suitable base region $B$ so that $\zeta (P({\mathscr A},B);\, t)$ admits a multiplicative decomposition which is determined by the exponents of $\mathscr A$ , that is

(6.1) \begin{equation} \zeta (P({\mathscr A},B);\, t) = \prod _{i=1}^\ell (1 + t + \ldots + t^{e_i}), \end{equation}

where $\{e_1, \ldots, e_\ell \} = \exp {\mathscr A}$ is the set of exponents of $\mathscr A$ .

Quite remarkably, many classical real arrangements do satisfy the factorization identity (6.1), the most prominent ones being Coxeter arrangements.

Let $W = (W,S)$ be a Coxeter group with associated reflection arrangement ${\mathscr A} = {\mathscr A}{\kern1.5pt}(W)$ which consists of the reflecting hyperplanes of the reflections in $W$ in the real space $V={\mathbb R}^n$ , where $|S| = n$ . The Poincaré polynomial $W(t)$ of the Coxeter group $W$ is the polynomial in ${\mathbb Z}[t]$ defined by

(6.2) \begin{equation} W(t)\, :\!= \sum _{w \in W} t^{\ell (w)}, \end{equation}

where $\ell$ is the length function of $W$ with respect to $S$ . Then $W(t)$ coincides with the rank-generating function of the poset of regions $\zeta (P({\mathscr A}{\kern2pt},B);\, t)$ of the underlying reflection arrangement ${\mathscr A}{\kern2pt} = {\mathscr A}{\kern1.5pt}(W)$ with respect to $B$ being the dominant Weyl chamber of $W$ in $V$ ; see [Reference Björner, Edelman and Ziegler8] or [Reference Jambu and Paris27].

The following factorization of $W(t)$ is due to Solomon [Reference Solomon40]:

(6.3) \begin{equation} W(t) = \prod _{i=1}^n(1 + t + \ldots + t^{e_i}), \end{equation}

where $\{e_1, \ldots, e_n\}$ is the set of exponents of $W$ , that is, the set of exponents of ${\mathscr A}{\kern1.5pt}(W)$ ; see also [Reference Macdonald29]. So by the comments above, (6.3) coincides with the factorization in (6.1).

Let $W$ be a Coxeter group again with reflection arrangement ${\mathscr A} = {\mathscr A}{\kern1.5pt}(W)$ , let $X$ be a member of the intersection lattice $L({\mathscr A}{\kern1.5pt})$ , and consider the restricted reflection arrangement ${\mathscr A}^{{\kern1.5pt}X}$ . In general, ${\mathscr A}^{{\kern1.5pt}X}$ is no longer a reflection arrangement. It was shown in [Reference Möller and Röhrle32, Thm. 1.3] that there always exists a suitable base region $B$ of ${\mathscr A}^{{\kern1.5pt}X}$ in $X$ so that also $\zeta (P({\mathscr A}^{{\kern1.5pt}X},B);\, t)$ satisfies (6.1) provided $W$ is not of type $E_8$ . In case $W$ is of type $E_8$ , then $\zeta (P({\mathscr A}^{{\kern1.5pt}X},B);\, t)$ satisfies (6.1) provided $X$ has rank at most 3 with only two exceptions or ${\mathscr A}^{{\kern1.5pt}X} \cong (E_8,D_4)$

Moreover, also the rank-generating function of the poset of regions $\zeta (P({\mathscr A}_{\mathcal I},B);\, t)$ for a so-called ideal arrangement ${\mathscr A}_{\mathcal I}$ also obeys the factorization identity (6.1). Ideal arrangements ${\mathscr A}_{\mathcal I}$ stem from ideals $\mathcal I$ in the poset of positive roots associated with a Weyl group; see [Reference Sommers and Tymoczko41], [Reference Abe, Barakat, Cuntz, Hoge and Terao1], [Reference Röhrle38], and [Reference Abe, Horiguchi, Masuda, Murai and Sato3].

We close with a comment on the rank-generating function of the poset of regions of the free hyperpolygonal arrangements ${\mathscr H}_n$ .