# Inductive Freeness of Ziegler’s Canonical Multiderivations for Restrictions
of Reflection Arrangements

Torsten Hoge, Gerhard Röhrle, and Sven Wiesner  Fakultät für Mathematik, Ruhr-
Universität Bochum, D-44780 Bochum, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let ${\mathscr{A}}$ be a free hyperplane arrangement. In 1989, Ziegler showed
that the restriction ${\mathscr{A}}^{\prime\prime}$ of ${\mathscr{A}}$ to any
hyperplane endowed with the natural multiplicity $\kappa$ is then a free
multiarrangement. Recently, in [HR22], an analogue of Ziegler’s theorem for
the stronger notion of inductive freeness was proved: if ${\mathscr{A}}$ is
inductively free, then so is the free multiarrangement
$({\mathscr{A}}^{\prime\prime},\kappa)$.

In [HR18], all reflection arrangements which admit inductively free Ziegler
restrictions are classified. The aim of this paper is an extension of this
classification to all restrictions of reflection arrangements utilizing the
aforementioned fundamental result from [HR22].

###### Key words and phrases:

Free arrangement, free multiarrangement, Ziegler multiplicity, inductively
free arrangement, reflection arrangement, restrictions of reflection
arrangements

###### 2010 Mathematics Subject Classification:

52C35 (14N20, 32S22, 51D20)

##  1\. Introduction

The class of free arrangements, respectively free multiarrangments, plays a
fundamental role in the theory of hyperplane arrangements, respectively
multiarrangements. In [Z89], Ziegler introduced the notion of
multiarrangements and initiated the study of their freeness. We begin by
recalling Ziegler’s fundamental construction from _loc. cit_.

######  Definition 1.1.

Let ${\mathscr{A}}$ be an arrangement. Fix $H_{0}\in{\mathscr{A}}$ and
consider the restriction ${\mathscr{A}}^{\prime\prime}$ with respect to
$H_{0}$. Define the _canonical multiplicity_ $\kappa$ on
${\mathscr{A}}^{\prime\prime}$ as follows. For
$Y\in{\mathscr{A}}^{\prime\prime}$ set

$\kappa(Y):=|{\mathscr{A}}_{Y}|-1,$

i.e., $\kappa(Y)$ is the number of hyperplanes in
${\mathscr{A}}\setminus\\{H_{0}\\}$ lying above $Y$. Ziegler showed that
freeness of ${\mathscr{A}}$ implies freeness of the multiarrangement
$({\mathscr{A}}^{\prime\prime},\kappa)$ as follows.

######  Theorem 1.2 ([Z89, Thm. 11]).

Let ${\mathscr{A}}$ be a free arrangement with exponents
$\exp{\mathscr{A}}=\\{1,e_{2},\ldots,e_{\ell}\\}$. Let $H_{0}\in{\mathscr{A}}$
and consider the restriction ${\mathscr{A}}^{\prime\prime}$ with respect to
$H_{0}$. Then the multiarrangement $({\mathscr{A}}^{\prime\prime},\kappa)$ is
free with exponents
$\exp({\mathscr{A}}^{\prime\prime},\kappa)=\\{e_{2},\ldots,e_{\ell}\\}$.

Because of the relevance of Ziegler’s multiplicity in the theory of free
arrangements, it is natural to investigate stronger freeness properties for
$({\mathscr{A}}^{\prime\prime},\kappa)$ and specifically to ask for an
analogue of Theorem 1.2 for inductive freeness. This was settled recently in
[HR22]:

######  Theorem 1.3 ([HR22, Thm. 1.3]).

If ${\mathscr{A}}$ is inductively free, then so is the Ziegler restriction
$({\mathscr{A}}^{H_{0}},\kappa)$, for every $H_{0}\in{\mathscr{A}}$.

Even if ${\mathscr{A}}$ itself is not inductively free, the following result
from [HR22] still guarantees inductive freeness of
$({\mathscr{A}}^{H_{0}},\kappa)$, provided the restriction
${\mathscr{A}}^{H_{0}}$ itself is inductively free.

######  Theorem 1.4 ([HR22, Thm. 1.6]).

Let $H_{0}\in{\mathscr{A}}$. Suppose that ${\mathscr{A}}\setminus\\{H_{0}\\}$
is free. If ${\mathscr{A}}^{H_{0}}$ is inductively free, then so is
$({\mathscr{A}}^{H_{0}},\kappa)$.

We note that here the freeness assumption on
${\mathscr{A}}\setminus\\{H_{0}\\}$ is needed, cf. [HR22, Ex. 4.2].

Because of the compatibility of the product constructions for inductive
freeness for simple arrangements, [HR15, Prop. 2.10], as well as for
multiarrangements, [HRS17, Thm. 1.4] (cf. Theorem 2.13), the question of
inductive freeness of $({\mathscr{A}}^{\prime\prime},\kappa)$ readily reduces
to the case when ${\mathscr{A}}$ is irreducible.

In [HR18], we classified all reflection arrangements ${\mathscr{A}}(W)$ which
admit Ziegler restrictions $({\mathscr{A}}(W)^{\prime\prime},\kappa)$ that are
inductively free.

######  Theorem 1.5 ([HR18, Thm. 1.4]).

Let ${\mathscr{A}}={\mathscr{A}}(W)$ be the reflection arrangement of the
irreducible complex reflection group $W$. Let ${\mathscr{A}}^{\prime\prime}$
be the restriction of ${\mathscr{A}}$ to a hyperplane in ${\mathscr{A}}$. Then
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free if and only if one
of the following holds:

  * (i)

${\mathscr{A}}$ is inductively free; or

  * (ii)

${\mathscr{A}}$ is non-inductively free of rank at most $4$.

We record a consequence of Theorem 1.5 along with the classification of all
inductively free restrictions of reflection arrangements from [AHR14, Thm.
1.2], (cf. Theorem 4.3).

######  Corollary 1.6 ([HR18, Cor. 1.5]).

Let ${\mathscr{A}}={\mathscr{A}}(W)$ be the reflection arrangement of the
complex reflection group $W$. Then $({\mathscr{A}}^{\prime\prime},\kappa)$ is
inductively free if and only if ${\mathscr{A}}^{\prime\prime}$ itself is
inductively free.

The goal of this paper is to extend the classification of Theorem 1.5 and
Corollary 1.6 to restrictions of complex reflection arrangements. Clearly, the
question reduces readily to the case when $W$ is irreducible. Thanks to work
by Orlik–Terao [OT93], [OT92, Prop. 6.73, Prop. 6.77, Cor. 6.86, App. D] and
Hoge–Röhrle [HR13], every restriction ${\mathscr{A}}(W)^{X}$ of a reflection
arrangement ${\mathscr{A}}(W)$ is free for $W$ an irreducible complex
reflection group. So in particular, the Ziegler multiplicity of a restriction
of the latter to a hyperplane is free, by Theorem 1.2. Here is the principal
result of our paper.

######  Theorem 1.7.

Let $W$ be a finite, irreducible, complex reflection group with reflection
arrangement ${\mathscr{A}}(W)$. Fix $X\in L({\mathscr{A}}(W))\setminus\\{V\\}$
and let ${\mathscr{A}}={\mathscr{A}}(W)^{X}$ be the restriction of
${\mathscr{A}}(W)$ to $X$. Let ${\mathscr{A}}^{\prime\prime}$ be the
restriction of ${\mathscr{A}}$ to a hyperplane in ${\mathscr{A}}$. Then
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free if and only if one
of the following holds:

  * (i)

${\mathscr{A}}(W)$ is inductively free; or

  * (ii)

$W=G(r,r,\ell)$ and the restriction ${\mathscr{A}}^{\prime\prime}$ of
${\mathscr{A}}$ itself is inductively free; or

  * (iii)

$W$ is of exceptional type $G_{33}$ or $G_{34}$ and
$\operatorname{rank}{\mathscr{A}}\leq 4$.

The reverse implication in part (i) of Theorem 1.7 follows from Theorem 1.3
and the fact that ${\mathscr{A}}(W)$ is inductively free if and only if it is
hereditarily inductively free, thanks to [HR15, Thm. 1.2]; see Theorem
4.1(ii).

Theorem 1.5 shows that the question of inductive freeness of
$({\mathscr{A}}^{\prime\prime},\kappa)$ for ${\mathscr{A}}={\mathscr{A}}(W)$ a
complex reflection arrangement does not depend on the choice of hyperplane. In
contrast for restrictions of complex reflection arrangements this question is
sensitive to the choice of hyperplane for the restriction, according to
Theorem 1.7(ii) in view of Theorem 4.3(i)(b).

We record several consequences of Theorem 1.7. Our first observation follows
from the theorem along with the classification of all inductively free
restrictions of reflection arrangements from [AHR14, Thm. 1.2], see Theorem
4.3 below, which is the analogue of Corollary 1.6 for the class of
restrictions of reflection arrangements.

######  Corollary 1.8.

Let $W$ be a finite, irreducible, complex reflection group with reflection
arrangement ${\mathscr{A}}(W)$ and let $X\in L({\mathscr{A}}(W))$. Let
${\mathscr{A}}={\mathscr{A}}(W)^{X}$ be the restriction of ${\mathscr{A}}(W)$
to $X$. Then $({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free if
and only if ${\mathscr{A}}^{\prime\prime}$ itself is inductively free.

We mention that the equivalences from both Corollaries 1.6 and 1.8 are false
in general. For the failure of the forward implication, see [HR22, Ex. 1.13]
and for the failure of the reverse implication, see [HR18, Ex. 2.24]. In view
of these elementary counterexamples, the equivalences of these two corollaries
are rather striking.

We also study multiplicities which are concentrated at a single hyperplane.
These were introduced by Abe, Terao and Wakefield, [ATW08, §5]. It turns out
that they are closely related to Ziegler’s canonical multiplicity, see
Proposition 3.1.

######  Definition 1.9.

Let ${\mathscr{A}}$ be a simple arrangement. Fix $H_{0}\in{\mathscr{A}}$ and
$m_{0}\in{\mathbb{Z}}_{\geq 1}$ and define the _multiplicity $\delta$
concentrated at $H_{0}$_ by

$\delta(H):=\delta_{H_{0},m_{0}}(H):=\begin{cases}m_{0}&\text{ if }H=H_{0},\\\
1&\text{ else}.\end{cases}$

In general a multiarrangement $({\mathscr{A}},\mu)$ need not be free for a
free hyperplane arrangement ${\mathscr{A}}$ and an arbitrary multiplicity
$\mu$, e.g. see [Z89, Ex. 14]. However, as opposed to the general case, both
${\mathscr{A}}$ and $({\mathscr{A}},\delta)$ do inherit freeness from each
other.

######  Theorem 1.10 ([HR18, Thm. 1.7]).

Let ${\mathscr{A}}$ be an arrangement. Fix $H_{0}\in{\mathscr{A}}$,
$m_{0}\in{\mathbb{Z}}_{\geq 1}$ and let $\delta=\delta_{H_{0},m_{0}}$ be the
multiplicity concentrated at $H_{0}$. Then ${\mathscr{A}}$ is free with
exponents $\exp{\mathscr{A}}=\\{1,e_{2},\ldots,e_{\ell}\\}$ if and only if
$({\mathscr{A}},\delta)$ is free with exponents
$\exp({\mathscr{A}},\delta)=\\{m_{0},e_{2},\ldots,e_{\ell}\\}$.

Our aim here is to strengthen Theorem 1.10 to inductive freeness.

######  Theorem 1.11.

Let $H_{0}\in{\mathscr{A}}$, $m_{0}\in{\mathbb{Z}}_{\geq 1}$ and let
$\delta=\delta_{H_{0},m_{0}}$ be the multiplicity concentrated at $H_{0}$.
Then ${\mathscr{A}}$ is inductively free if and only if
$({\mathscr{A}},\delta)$ is inductively free.

The forward implication of Theorem 1.11 is [HR22, Cor. 5.6]. We observe that
Theorem 1.11 gives a uniform case free proof of [HR18, Thm. 1.9].

The paper is organized as follows. In Section 2.1 we recall the fundamental
results for free arrangements, in particular Terao’s Addition Deletion Theorem
2.1 and the subsequent notion of an inductively free arrangement.

Section 2.2 is devoted to multiarrangements and their freeness. Here we
present the Addition Deletion Theorem due to Abe, Terao, and Wakefield,
[ATW08, Thm. 0.8], see Theorem 2.9. This is followed by a discussion of
inductive freeness for multiarrangements. Here we recall results from [HRS17]
which show the compatibility of this notion with products and localization for
multiarrangements that are used in the sequel.

In Section 4 we recall the classification of the inductively free reflection
arrangements from [HR15, Thm. 1.1, 1.2] (Theorem 4.1) and the classification
of the inductively free restrictions of reflection arrangements [AHR14, Thm.
1.2, Thm. 1.3] (Theorem 4.3). We combine the latter with Theorems 1.3 and 1.4
to obtain a first result towards Theorem 1.7 in Theorem 4.2.

Theorem 1.11 is proved in Section 3 and Theorem 1.7 in Section 5.

##  2\. Recollections and Preliminaries

###  2.1. Freeness of hyperplane arrangements

Let $V={\mathbb{K}}^{\ell}$ be an $\ell$-dimensional ${\mathbb{K}}$-vector
space. A _hyperplane arrangement_ is a pair $({\mathscr{A}},V)$, where
${\mathscr{A}}$ is a finite collection of hyperplanes in $V$. Usually, we
simply write ${\mathscr{A}}$ in place of $({\mathscr{A}},V)$. We write
$|{\mathscr{A}}|$ for the number of hyperplanes in ${\mathscr{A}}$. The empty
arrangement in $V$ is denoted by $\Phi_{\ell}$.

The _lattice_ $L({\mathscr{A}})$ of ${\mathscr{A}}$ is the set of subspaces of
$V$ of the form $H_{1}\cap\dotsm\cap H_{i}$ where $\\{H_{1},\ldots,H_{i}\\}$
is a subset of ${\mathscr{A}}$. For $X\in L({\mathscr{A}})$, we have two
associated arrangements, firstly ${\mathscr{A}}_{X}:=\\{H\in{\mathscr{A}}\mid
X\subseteq H\\}\subseteq{\mathscr{A}}$, the _localization of ${\mathscr{A}}$
at $X$_, and secondly, the _restriction of ${\mathscr{A}}$ to $X$_,
$({\mathscr{A}}^{X},X)$, where ${\mathscr{A}}^{X}:=\\{X\cap H\mid
H\in{\mathscr{A}}\setminus{\mathscr{A}}_{X}\\}$. Note that $V$ belongs to
$L({\mathscr{A}})$ as the intersection of the empty collection of hyperplanes
and ${\mathscr{A}}^{V}={\mathscr{A}}$. The lattice $L({\mathscr{A}})$ is a
partially ordered set by reverse inclusion: $X\leq Y$ provided $Y\subseteq X$
for $X,Y\in L({\mathscr{A}})$.

Suppose ${\mathscr{A}}\neq\Phi_{\ell}$. Fix a member $H_{0}$ in
${\mathscr{A}}$. Set
${\mathscr{A}}^{\prime}={\mathscr{A}}\setminus\\{H_{0}\\}$ and
${\mathscr{A}}^{\prime\prime}={\mathscr{A}}^{H_{0}}$. Then
$({\mathscr{A}},{\mathscr{A}}^{\prime},{\mathscr{A}}^{\prime\prime})$ is
frequently referred to as a _triple of arrangements_.

Let $S=S(V^{*})$ be the symmetric algebra of the dual space $V^{*}$ of $V$. If
$x_{1},\ldots,x_{\ell}$ is a basis of $V^{*}$, then we identify $S$ with the
polynomial ring ${\mathbb{K}}[x_{1},\ldots,x_{\ell}]$. Letting $S_{p}$ denote
the ${\mathbb{K}}$-subspace of $S$ consisting of the homogeneous polynomials
of degree $p$ (along with $0$), $S$ is naturally ${\mathbb{Z}}$-graded:
$S=\oplus_{p\in{\mathbb{Z}}}S_{p}$, where $S_{p}=0$ in case $p<0$.

Let ${\operatorname{Der}}(S)$ be the $S$-module of algebraic
${\mathbb{K}}$-derivations of $S$. Using the ${\mathbb{Z}}$-grading on $S$,
${\operatorname{Der}}(S)$ becomes a graded $S$-module. For $i=1,\ldots,\ell$,
let $D_{i}:=\partial/\partial x_{i}$ be the usual derivation of $S$. Then
$D_{1},\ldots,D_{\ell}$ is an $S$-basis of ${\operatorname{Der}}(S)$. We say
that $\theta\in{\operatorname{Der}}(S)$ is _homogeneous of polynomial degree
p_ provided $\theta=\sum_{i=1}^{\ell}f_{i}D_{i}$, where $f_{i}$ is either $0$
or homogeneous of degree $p$ for each $1\leq i\leq\ell$. In this case we write
$\operatorname{pdeg}\theta=p$.

Let ${\mathscr{A}}$ be an arrangement in $V$. Then for $H\in{\mathscr{A}}$ we
fix $\alpha_{H}\in V^{*}$ with $H=\ker(\alpha_{H})$. The _defining polynomial_
$Q({\mathscr{A}})$ of ${\mathscr{A}}$ is given by
$Q({\mathscr{A}}):=\prod_{H\in{\mathscr{A}}}\alpha_{H}\in S$.

The _module of ${\mathscr{A}}$-derivations_ of ${\mathscr{A}}$ is defined by

$D({\mathscr{A}}):=\\{\theta\in{\operatorname{Der}}(S)\mid\theta(\alpha_{H})\in\alpha_{H}S\text{
for each }H\in{\mathscr{A}}\\}.$

We say that ${\mathscr{A}}$ is _free_ if the module of
${\mathscr{A}}$-derivations $D({\mathscr{A}})$ is a free $S$-module.

With the ${\mathbb{Z}}$-grading of ${\operatorname{Der}}(S)$, also
$D({\mathscr{A}})$ becomes a graded $S$-module, [OT92, Prop. 4.10]. If
${\mathscr{A}}$ is a free arrangement, then the $S$-module $D({\mathscr{A}})$
admits a basis of $\ell$ homogeneous derivations, say
$\theta_{1},\ldots,\theta_{\ell}$, [OT92, Prop. 4.18]. While the
$\theta_{i}$’s are not unique, their polynomial degrees
$\operatorname{pdeg}\theta_{i}$ are unique (up to ordering). This multiset is
the set of _exponents_ of the free arrangement ${\mathscr{A}}$ and is denoted
by $\exp{\mathscr{A}}$.

The fundamental _Addition Deletion Theorem_ due to Terao [T80a] plays a
crucial role in the study of free arrangements, [OT92, Thm. 4.51].

######  Theorem 2.1.

Suppose ${\mathscr{A}}\neq\Phi_{\ell}$ and let
$({\mathscr{A}},{\mathscr{A}}^{\prime},{\mathscr{A}}^{\prime\prime})$ be a
triple of arrangements. Then any two of the following statements imply the
third:

  * (i)

${\mathscr{A}}$ is free with
$\exp{\mathscr{A}}=\\{b_{1},\ldots,b_{\ell-1},b_{\ell}\\}$;

  * (ii)

${\mathscr{A}}^{\prime}$ is free with
$\exp{\mathscr{A}}^{\prime}=\\{b_{1},\ldots,b_{\ell-1},b_{\ell}-1\\}$;

  * (iii)

${\mathscr{A}}^{\prime\prime}$ is free with
$\exp{\mathscr{A}}^{\prime\prime}=\\{b_{1},\ldots,b_{\ell-1}\\}$.

Theorem 2.1 motivates the following notion.

######  Definition 2.2 ([OT92, Def. 4.53]).

The class ${\mathcal{IF}}$ of _inductively free_ arrangements is the smallest
class of arrangements subject to

  * (i)

$\Phi_{\ell}\in{\mathcal{IF}}$ for each $\ell\geq 0$;

  * (ii)

if there exists a hyperplane $H_{0}\in{\mathscr{A}}$ such that both
${\mathscr{A}}^{\prime}$ and ${\mathscr{A}}^{\prime\prime}$ belong to
${\mathcal{IF}}$, and
$\exp{\mathscr{A}}^{\prime\prime}\subseteq\exp{\mathscr{A}}^{\prime}$, then
${\mathscr{A}}$ also belongs to ${\mathcal{IF}}$.

######  Definition 2.3 ([OT92, p. 253]).

An arrangement ${\mathscr{A}}$ is called _hereditarily inductively free_ if
${\mathscr{A}}$ and every restriction of ${\mathscr{A}}$ is inductively free.

###  2.2. Freeness of multiarrangements

A _multiarrangement_ is a pair $({\mathscr{A}},\mu)$ consisting of a
hyperplane arrangement ${\mathscr{A}}$ and a _multiplicity_ function
$\mu:{\mathscr{A}}\to{\mathbb{Z}}_{\geq 0}$ associating to each hyperplane $H$
in ${\mathscr{A}}$ a non-negative integer $\mu(H)$. The _order_ of the
multiarrangement $({\mathscr{A}},\mu)$ is defined by
$|\mu|:=\sum_{H\in{\mathscr{A}}}\mu(H)$. For a multiarrangement
$({\mathscr{A}},\mu)$, the underlying arrangement ${\mathscr{A}}$ is sometimes
called the associated _simple_ arrangement, and so $({\mathscr{A}},\mu)$
itself is simple if and only if $\mu(H)=1$ for each $H\in{\mathscr{A}}$. We
indicate this case with the notation $\mu=\mathbbm{1}$.

Let $({\mathscr{A}},\mu)$ be a multiarrangement in $V$ and let $X\in
L({\mathscr{A}})$. The _localization of $({\mathscr{A}},\mu)$ at $X$_ is
defined to be $({\mathscr{A}}_{X},\mu_{X})$, where
$\mu_{X}=\mu|_{{\mathscr{A}}_{X}}$.

The _defining polynomial_ $Q({\mathscr{A}},\mu)$ of the multiarrangement
$({\mathscr{A}},\mu)$ is given by

$Q({\mathscr{A}},\mu):=\prod_{H\in{\mathscr{A}}}\alpha_{H}^{\mu(H)},$

a polynomial of degree $|\mu|$ in $S$.

Following Ziegler [Z89], we extend the notion of freeness to multiarrangements
as follows. The _module of ${\mathscr{A}}$-derivations_ of
$({\mathscr{A}},\mu)$ is defined by

$D({\mathscr{A}},\mu):=\\{\theta\in{\operatorname{Der}}(S)\mid\theta(\alpha_{H})\in\alpha_{H}^{\mu(H)}S\text{
for each }H\in{\mathscr{A}}\\}.$

We say that $({\mathscr{A}},\mu)$ is _free_ if $D({\mathscr{A}},\mu)$ is a
free $S$-module, [Z89, Def. 6].

As in the case of simple arrangements, $D({\mathscr{A}},\mu)$ is a
${\mathbb{Z}}$-graded $S$-module and thus, if $({\mathscr{A}},\mu)$ is free,
there is a homogeneous basis $\theta_{1},\ldots,\theta_{\ell}$ of
$D({\mathscr{A}},\mu)$. The multiset of the unique polynomial degrees
$\operatorname{pdeg}\theta_{i}$ forms the set of _exponents_ of the free
multiarrangement $({\mathscr{A}},\mu)$ and is denoted by
$\exp({\mathscr{A}},\mu)$.

We recall the construction from [ATW08] for the counterpart of Theorem 2.1 in
this more general setting.

######  Definition 2.4.

Let $({\mathscr{A}},\mu)\neq\Phi_{\ell}$ be a multiarrangement. Fix $H_{0}$ in
${\mathscr{A}}$. We define the _deletion_
$({\mathscr{A}}^{\prime},\mu^{\prime})$ and _Euler restriction_
$({\mathscr{A}}^{\prime\prime},\mu^{*})$ of $({\mathscr{A}},\mu)$ with respect
to $H_{0}$ as follows. If $\mu(H_{0})=1$, then set
${\mathscr{A}}^{\prime}={\mathscr{A}}\setminus\\{H_{0}\\}$ and define
$\mu^{\prime}(H)=\mu(H)$ for all $H\in{\mathscr{A}}^{\prime}$. If
$\mu(H_{0})>1$, then set ${\mathscr{A}}^{\prime}={\mathscr{A}}$ and define
$\mu^{\prime}(H_{0})=\mu(H_{0})-1$ and $\mu^{\prime}(H)=\mu(H)$ for all $H\neq
H_{0}$.

Let ${\mathscr{A}}^{\prime\prime}=\\{H\cap H_{0}\mid
H\in{\mathscr{A}}\setminus\\{H_{0}\\}\ \\}$. The _Euler multiplicity_
$\mu^{*}$ of ${\mathscr{A}}^{\prime\prime}$ is defined as follows. Let
$Y\in{\mathscr{A}}^{\prime\prime}$. Since the localization ${\mathscr{A}}_{Y}$
is of rank $2$, the multiarrangement $({\mathscr{A}}_{Y},\mu_{Y})$ is free,
[Z89, Cor. 7]. According to [ATW08, Prop. 2.1], the module of derivations
$D({\mathscr{A}}_{Y},\mu_{Y})$ admits a particular homogeneous basis
$\\{\theta_{Y},\psi_{Y},D_{3},\ldots,D_{\ell}\\}$, such that
$\theta_{Y}\notin\alpha_{0}{\operatorname{Der}}(S)$ and
$\psi_{Y}\in\alpha_{0}{\operatorname{Der}}(S)$, where $H_{0}=\ker\alpha_{0}$.
Then on $Y$ the Euler multiplicity $\mu^{*}$ is defined to be
$\mu^{*}(Y)=\operatorname{pdeg}\theta_{Y}$.

Often, we refer to $({\mathscr{A}},\mu),({\mathscr{A}}^{\prime},\mu^{\prime})$
and $({\mathscr{A}}^{\prime\prime},\mu^{*})$ as the _triple_ of
$({\mathscr{A}},\mu)$ with respect to $H_{0}$.

The following observation is a direct consequence of Definition 2.4.

######  Remark 2.5.

Let $({\mathscr{A}},\mu)$ be a multiarrangement, $H_{0}\in{\mathscr{A}}$, and
let $({\mathscr{A}}^{\prime\prime},\mu^{*})$ be the Euler restriction
corresponding to $H_{0}$. For $X\in{\mathscr{A}}^{\prime\prime}$ the value
$\mu^{*}(X)$ is simply the common value of the non-zero exponents of
$({\mathscr{A}}_{X},\mu_{X})$ and
$({\mathscr{A}}_{X}^{\prime},\mu_{X}^{\prime})$.

It is useful to be able to determine the Euler multiplicity explicitly in some
relevant instances. For that purpose, we recall parts of [ATW08, Prop. 4.1].

######  Proposition 2.6.

Let $({\mathscr{A}},\mu)$ be a multiarrangement. Let
$X\in{\mathscr{A}}^{H_{0}}$, $m_{0}=\mu(H_{0})$, $k=|{\mathscr{A}}_{X}|$, and
$m_{1}=\max\\{\mu(H)\mid H\in{\mathscr{A}}_{X}\backslash\\{H_{0}\\}\\}$.

  1. (1)

If $k=2$, then $\mu^{*}(X)=m_{1}$.

  2. (2)

If $|\mu_{X}|\leq 2k-1$ and $m_{0}>1$, then $\mu^{*}(X)=k-1$.

  3. (3)

If $\mu_{X}\equiv 2$, then $\mu^{*}(X)=k$.

The following is a special case of a general compatibility result of Euler
restrictions with localizations from [HRS17, Lem. 2.14].

######  Lemma 2.7 ([HR18, Lem. 2.16]).

Let $X\in L({\mathscr{A}})$, $H_{0}\in{\mathscr{A}}_{X}$, and let
${\mathscr{A}}^{\prime\prime}$ be the restriction with respect to $H_{0}$. Let
$\delta=\delta_{H_{0},m_{0}}$ be as in Definition 1.9. Then we have

  * (i)

$\left(({\mathscr{A}}_{X})^{\prime\prime},(\delta_{X})^{*}\right)=\left(({\mathscr{A}}^{\prime\prime})_{X},(\delta^{*})_{X}\right)$,
and

  * (ii)

$\left(({\mathscr{A}}_{X})^{\prime\prime},\kappa\right)=\left(({\mathscr{A}}^{\prime\prime})_{X},\kappa_{X}\right)$
(where $\kappa$ on the left is the canonical multiplicity resulting from
restriction of ${\mathscr{A}}_{X}$ to $H_{0}$).

We also require the following tool from [HR22]. This result turns out to be
useful in our sequel in determining inductively free Ziegler restrictions.

######  Lemma 2.8 ([HR22, Lem. 2.12]).

Let ${\mathscr{A}}$ be an $\ell$-arrangement and let $H_{1}$,…,$H_{n}$ be
distinct hyperplanes in ${\mathscr{A}}$. Define
${\mathscr{A}}_{0}:={\mathscr{A}}\setminus\\{H_{1},\ldots,H_{n}\\}$, and
${\mathscr{A}}_{i}:={\mathscr{A}}_{i-1}\cup\\{H_{i}\\}$ for $i=1,\ldots,n$.
Suppose that ${\mathscr{A}}_{0},\ldots,{\mathscr{A}}_{n}={\mathscr{A}}$ are
all free. Then for a fixed $H\in{\mathscr{A}}_{0}$, we have the following:

  1. (1)

$({\mathscr{A}}_{i}^{H},\kappa_{i})$ is free for $i=0,\ldots,n$, where
$\kappa_{i}$ is the Ziegler multiplicity of ${\mathscr{A}}_{i}^{H}$.

  2. (2)

Consider the triple $({\mathscr{A}}_{i}^{H},\kappa_{i})$,
$({\mathscr{A}}_{i-1}^{H},\kappa_{i-1})$, and $(({\mathscr{A}}_{i}^{H})^{H\cap
H_{i}},\kappa_{i}^{*})$. Then the Euler multiplicity $\kappa_{i}^{*}$ is just
the Ziegler multiplicity $\kappa$ of the restriction of
${\mathscr{A}}_{i}^{H_{i}}$ to $H\cap H_{i}$.

Here is the counterpart to Theorem 2.1 in this more general setting from
[ATW08].

######  Theorem 2.9 ([ATW08, Thm. 0.8] Addition Deletion Theorem for
Multiarrangements).

Suppose that $({\mathscr{A}},\mu)\neq\Phi_{\ell}$. Fix $H_{0}$ in
${\mathscr{A}}$ and let
$({\mathscr{A}},\mu),({\mathscr{A}}^{\prime},\mu^{\prime})$ and
$({\mathscr{A}}^{\prime\prime},\mu^{*})$ be the triple with respect to
$H_{0}$. Then any two of the following statements imply the third:

  * (i)

$({\mathscr{A}},\mu)$ is free with
$\exp({\mathscr{A}},\mu)=\\{b_{1},\ldots,b_{\ell-1},b_{\ell}\\}$;

  * (ii)

$({\mathscr{A}}^{\prime},\mu^{\prime})$ is free with
$\exp({\mathscr{A}}^{\prime},\mu^{\prime})=\\{b_{1},\ldots,b_{\ell-1},b_{\ell}-1\\}$;

  * (iii)

$({\mathscr{A}}^{\prime\prime},\mu^{*})$ is free with
$\exp({\mathscr{A}}^{\prime\prime},\mu^{*})=\\{b_{1},\ldots,b_{\ell-1}\\}$.

Analogous to the simple case, Theorem 2.9 motivates the notion of inductive
freeness.

######  Definition 2.10 ([ATW08, Def. 0.9]).

The class ${\mathcal{IFM}}$ of _inductively free_ multiarrangements is the
smallest class of multiarrangements subject to

  * (i)

$\Phi_{\ell}\in{\mathcal{IFM}}$ for each $\ell\geq 0$;

  * (ii)

for a multiarrangement $({\mathscr{A}},\mu)$, if there exists a hyperplane
$H_{0}\in{\mathscr{A}}$ such that both $({\mathscr{A}}^{\prime},\mu^{\prime})$
and $({\mathscr{A}}^{\prime\prime},\mu^{*})$ belong to ${\mathcal{IFM}}$, and
$\exp({\mathscr{A}}^{\prime\prime},\mu^{*})\subseteq\exp({\mathscr{A}}^{\prime},\mu^{\prime})$,
then $({\mathscr{A}},\mu)$ also belongs to ${\mathcal{IFM}}$.

######  Remark 2.11.

As for simple arrangements, if $r({\mathscr{A}})\leq 2$, then
$({\mathscr{A}},\mu)$ is inductively free for any multipliity $\mu$, [Z89,
Cor. 7].

The following is a useful tool for showing that a given multiarrangement is
not inductively free by exhibiting a localization which fails to be
inductively free.

######  Theorem 2.12 ([HRS17, Thm. 1.3]).

The class ${\mathcal{IFM}}$ is closed under taking localizations.

We also require the fact that inductive freeness for multiarrangements behaves
well with the product construction.

######  Theorem 2.13 ([HRS17, Thm. 1.4]).

A product of multiarrangements belongs to ${\mathcal{IFM}}$ if and only if
each factor belongs to ${\mathcal{IFM}}$.

We recall a natural partial order on the set of multiplicities for a simple
arrangement and the notion of an additively free multiarrangement from [HR22,
Def. 2.19].

######  Definition 2.14.

Let ${\mathscr{A}}$ be a fixed $\ell$-arrangement.

  * (i)

For multiplicities $\mu_{1}$ and $\mu_{2}$ on ${\mathscr{A}}$, define
$\mu_{1}\leq\mu_{2}$ provided $\mu_{1}(H)\leq\mu_{2}(H)$ for every $H$ in
${\mathscr{A}}$.

  * (ii)

The multiarrangement $({\mathscr{A}},\mu)$ is said to be _additively free_ if
there is a _free filtration_ of multiplicities
$\mu_{0}<\mu_{1}<\cdots<\mu_{n}=\mu$, of $({\mathscr{A}},\mu)$, i.e., where
each $({\mathscr{A}},\mu_{i})$ is free with $|\mu_{i}|=i$. In particular,
$({\mathscr{A}},\mu_{0})=\Phi_{\ell}$.

Clearly, if $({\mathscr{A}},\mu)$ is inductively free, it is additively free.

##  3\. Proof of Theorem 1.11

Recall the notion of a concentrated multiplicity $\delta$ from Definition 1.9.
The following combines [ATW08, Prop. 5.2], parts of its proof and Theorem 1.2.
Wile not needed for the proof of Theorem 1.11, this result does illustrate the
connection with the canonical multiplicity.

######  Proposition 3.1 ([HR18, Prop. 2.14]).

Let ${\mathscr{A}}$ be a free arrangement with exponents
$\exp{\mathscr{A}}=\\{1,e_{2},\ldots,e_{\ell}\\}$. Let
$H_{0}\in{\mathscr{A}}$, $m_{0}\geq 1$, and $\delta=\delta_{H_{0},m_{0}}$. Let
$({\mathscr{A}}^{\prime\prime},\delta^{*})$ be the restriction with respect to
$H_{0}$. Then we have

  * (i)

$({\mathscr{A}},\delta)$ is free with exponents
$\exp({\mathscr{A}},\delta)=\\{m_{0},e_{2},\ldots,e_{\ell}\\}$;

  * (ii)

$({\mathscr{A}}^{\prime\prime},\delta^{*})=({\mathscr{A}}^{\prime\prime},\kappa)$
is free with exponents
$\exp({\mathscr{A}}^{\prime\prime},\kappa)=\\{e_{2},\ldots,e_{\ell}\\}$.

We require this basic observation.

######  Lemma 3.2.

Let $H_{0}\in{\mathscr{A}}$, $m_{0}\geq 1$, and $\delta=\delta_{H_{0},m_{0}}$.
Fix $H\in{\mathscr{A}}\setminus\\{H_{0}\\}$. Then
$({\mathscr{A}}^{H},\delta^{*})=({\mathscr{A}}^{H},\delta_{H_{0}\cap
H,m_{0}})$, i.e. $\delta^{*}$ is itself a concentrated multiplicity on
${\mathscr{A}}^{H}$.

###### Proof.

Let $X\in{\mathscr{A}}^{H}$. If $H_{0}\notin{\mathscr{A}}_{X}$, then
$\delta_{X}\equiv\mathbbm{1}$, so $\delta^{*}(X)=1$. If
$H_{0}\in{\mathscr{A}}_{X}$, then
$({\mathscr{A}}_{X},\delta_{X})=({\mathscr{A}}_{X},\delta_{H_{0},m_{0}})$ with
$\exp({\mathscr{A}}_{X},\delta_{X})=\\{m_{0},e\\}$ and
$\exp({\mathscr{A}}_{X}\setminus\\{H\\},\delta_{X}^{\prime})=\\{m_{0},e-1\\}$.
Thus, $\delta^{*}(X)=m_{0}$, by Remark 2.5. ∎

###### Proof of Theorem 1.11.

The forward implication is [HR22, Cor. 5.6].

For the reverse implication assume that $({\mathscr{A}},\delta)$ is
inductively free. We argue by induction on $|{\mathscr{A}}|$. If
$|{\mathscr{A}}|\leq 3$, then ${\mathscr{A}}$ itself is inductively free. So
suppose that $|{\mathscr{A}}|>3$ and that the result holds for arrangements
with concentrated multiplicities with fewer hyperplanes.

We argue further by induction on $m_{0}\geq 1$. If $m_{0}=1$, there is nothing
to show. So assume $m_{0}>1$ and that the result holds for a concentrated
multiplicity on ${\mathscr{A}}$ with a lower parameter.

Since $({\mathscr{A}},\delta)$ is inductively free, there is a hyperplane
$H\in{\mathscr{A}}$ so that the triple
$({\mathscr{A}},\delta),({\mathscr{A}}^{\prime},\delta^{\prime})$ and
$({\mathscr{A}}^{\prime\prime},\delta^{*})$ with respect to $H$ consists of
inductively free members obeying the containment condition for the exponents
from Theorem 2.9, cf. Definition 2.10.

If $H=H_{0}$, then
$({\mathscr{A}}^{\prime},\delta^{\prime})=({\mathscr{A}},\delta_{H_{0},m_{0}-1})$,
and so ${\mathscr{A}}$ is inductively free, by induction on $m_{0}$.

If $H\neq H_{0}$, then $|{\mathscr{A}}^{\prime}|<|{\mathscr{A}}|$ and
$|{\mathscr{A}}^{\prime\prime}|<|{\mathscr{A}}|$. By Lemma 3.2, $\delta^{*}$
is again a concentrated multiplicity on ${\mathscr{A}}^{\prime\prime}$. So it
follows from induction on $|{\mathscr{A}}|$ that both ${\mathscr{A}}^{\prime}$
and ${\mathscr{A}}^{\prime\prime}$ are inductively free and thanks to Theorems
1.10, 2.1, 2.9 and Definitions 2.2 and 2.10, ${\mathscr{A}}$ is inductively
free. ∎

######  Remark 3.3.

In the reverse implication of Theorem 1.11, using Theorem 1.3 and Proposition
3.1, one can show that any inductive chain of $({\mathscr{A}},\delta)$
descends to one for ${\mathscr{A}}$.

##  4\. Reflection arrangements and their restrictions

Let $W$ be a complex reflection group in $\operatorname{GL}(V)$. The
_reflection arrangement_ ${\mathscr{A}}={\mathscr{A}}(W)$ of $W$ in $V$ is the
hyperplane arrangement consisting of the reflecting hyperplanes of the
elements in $W$ acting as reflections in $V$. If
${\mathscr{A}}={\mathscr{A}}(W)$ is a reflection arrangement of the complex
reflection group $W$, then ${\mathscr{A}}$ is free, thanks to work of Terao
[T80b]. Thus $({\mathscr{A}}^{\prime\prime},\kappa)$ is also free, by Theorem
1.2.

The irreducible finite complex reflection groups were classified by Shephard
and Todd, [ST54]. Using the classification and nomenclature of [ST54], we
recall the classification of all inductively free reflection arrangements
${\mathscr{A}}(W)$.

######  Theorem 4.1 ([HR15, Thm. 1.1, 1.2]).

For a finite complex reflection group $W$, let
${\mathscr{A}}={\mathscr{A}}(W)$ be its reflection arrangement. Then the
following hold:

  * (i)

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

  * (ii)

${\mathscr{A}}$ is inductively free if and only if ${\mathscr{A}}$ is
hereditarily inductively free.

Thanks to work by Orlik–Terao [OT93], [OT92, Prop. 6.73, Prop. 6.77, Cor.
6.86, App. D] and Hoge–Röhrle [HR13], for $W$ an irreducible complex
reflection group, every restriction ${\mathscr{A}}(W)^{X}$ of the reflection
arrangement ${\mathscr{A}}(W)$ is free. So in particular, the Ziegler
multiplicity of a restriction of the latter to a hyperplane is free, by
Theorem 1.2.

The following is immediate from Theorems 1.3 and 4.1 and gives the reverse
implication of part (i) of Theorem 1.7.

######  Theorem 4.2.

Let $W$ be a finite complex reflection group with reflection arrangement
${\mathscr{A}}(W)$, let $X\in L({\mathscr{A}}(W))$ and set
${\mathscr{A}}={\mathscr{A}}(W)^{X}$. If ${\mathscr{A}}(W)$ is inductively
free, then so is $({\mathscr{A}}^{\prime\prime},\kappa)$.

Next we recall the classification of all inductively free restrictions
${\mathscr{A}}(W)^{X}$. For the definition of the arrangements
${\mathscr{A}}^{k}_{p}(r)$, see §4.1.

######  Theorem 4.3 ([AHR14, Thm. 1.2, Thm. 1.3]).

Let $W$ be a finite, irreducible, complex reflection group with reflection
arrangement ${\mathscr{A}}={\mathscr{A}}(W)$ and let $X\in L({\mathscr{A}})$.
Then

  * (i)

${\mathscr{A}}^{X}$ is inductively free if and only if one of the following
holds:

    * (a)

${\mathscr{A}}$ is inductively free;

    * (b)

$W=G(r,r,\ell)$ and ${\mathscr{A}}^{X}\cong{\mathscr{A}}^{k}_{p}(r)$, where
$p=\dim X$ and $p-2\leq k\leq p$;

    * (c)

$W$ is one of $G_{24},G_{27},G_{29},G_{31},G_{33}$, or $G_{34}$ and $X\in
L({\mathscr{A}})\setminus\\{V\\}$ with $\dim X\leq 3$;

  * (ii)

${\mathscr{A}}^{X}$ is inductively free if and only if it is hereditarily
inductively free.

It is immediate from Theorem 1.3 that if ${\mathscr{A}}={\mathscr{A}}(W)^{X}$
is inductively free, for $X\in L({\mathscr{A}}(W))$, then so is
$({\mathscr{A}}^{\prime\prime},\kappa)$. Moreover, we have the following for
low rank restrictions.

######  Corollary 4.4.

Let $W$ be a finite, irreducible, complex reflection group with reflection
arrangement ${\mathscr{A}}(W)$ and let $V\neq X\in L({\mathscr{A}}(W))$ with
$\dim X\leq 3$. Let ${\mathscr{A}}:={\mathscr{A}}(W)^{X}$. Then
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free.

###### Proof.

Clearly, in each instance Remark 2.11 gives that
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free. It follows from
Theorems 4.1 and 4.3 that the restricted arrangement ${\mathscr{A}}$ is also
inductively free. (In case $W=G(r,r,\ell)$ with $r,\ell\geq 3$, we have
${\mathscr{A}}(W)^{X}\cong{\mathscr{A}}^{k}_{3}(r)$ for $1\leq k\leq 2$, cf.
§4.1.) ∎

###  4.1. Intermediate Arrangements

Orlik and Solomon defined intermediate arrangements
${\mathscr{A}}^{k}_{\ell}(r)$ in [OS82, §2] (cf. [OT92, §6.4]) which
interpolate between the reflection arrangements of $G(r,r,\ell)$ and
$G(r,1,\ell)$. These are the ones that show up as restrictions of the
reflection arrangement of $G(r,r,n)$, for some $n$, [OS82, Prop. 2.14] (cf.
[OT92, Prop. 6.84]), see also [AHR14, Ex. 3.2].

For $r,\ell\geq 2$ and $0\leq k\leq\ell$ the defining polynomial of
${\mathscr{A}}^{k}_{\ell}(r)$ is given by

$Q({\mathscr{A}}^{k}_{\ell}(r))=x_{1}\cdots
x_{k}\prod\limits_{\begin{subarray}{c}1\leq i<j\leq\ell\\\ 0\leq
n<r\end{subarray}}(x_{i}-\zeta^{n}x_{j}),$

where $\zeta$ is a primitive $r{{}^{\text{th}}}$ root of unity, so that
${\mathscr{A}}^{\ell}_{\ell}(r)={\mathscr{A}}(G(r,1,\ell))$ and
${\mathscr{A}}^{0}_{\ell}(r)={\mathscr{A}}(G(r,r,\ell))$. We recall [OS82,
Props. 2.11, 2.13] (cf. [OT92, Props. 6.82, 6.85]):

######  Proposition 4.5.

Let ${\mathscr{A}}={\mathscr{A}}^{k}_{\ell}(r)$.

  1. (i)

${\mathscr{A}}$ is free with
$\exp{\mathscr{A}}=\\{1,r+1,\ldots,(\ell-2)r+1,(\ell-1)r-\ell+k+1\\}$.

  2. (ii)

Let $H\in{\mathscr{A}}$. The type of ${\mathscr{A}}^{H}$ is given in Table 1.

$k$ | $\alpha_{H}$ | Type of ${\mathscr{A}}^{H}$
---|---|---
$0$ | arbitrary |  | ${\mathscr{A}}^{1}_{\ell-1}(r)$
$1,\ldots,\ell-1$ | $x_{i}-\zeta x_{j}$ | $1\leq i<j\leq k<\ell$ | ${\mathscr{A}}^{k-1}_{\ell-1}(r)$
$1,\ldots,\ell-1$ | $x_{i}-\zeta x_{j}$ | $1\leq i\leq k<j\leq\ell$ | ${\mathscr{A}}^{k}_{\ell-1}(r)$
$1,\ldots,\ell-1$ | $x_{i}-\zeta x_{j}$ | $1\leq k<i<j\leq\ell$ | ${\mathscr{A}}^{k+1}_{\ell-1}(r)$
$1,\ldots,\ell-1$ | $x_{i}$ | $1\leq i\leq\ell$ | ${\mathscr{A}}^{\ell-1}_{\ell-1}(r)$
$\ell$ | arbitrary |  | ${\mathscr{A}}^{\ell-1}_{\ell-1}(r)$

Table 1. Restriction types of ${\mathscr{A}}^{k}_{\ell}(r)$

##  5\. Proof of Theorem 1.7

In order to classify all restrictions of reflection arrangements
${\mathscr{A}}={\mathscr{A}}(W)^{X}$ so that
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free, it follows from
Theorems 1.3 and 4.2 that we only need to investigate the instances when
either ${\mathscr{A}}(G(r,r,n))^{X}\cong{\mathscr{A}}^{k}_{\ell}(r)$, where
$\ell=\dim X$ and $1\leq k\leq\ell-3$, or when $W$ is of type $G_{33}$, or
$G_{34}$ and $X\in L({\mathscr{A}}(W))\setminus\\{V\\}$ with $\dim X\geq 4$.
We consider these instances separately.

Theorem 1.7 and Corollary 1.8 follow from the subsequent lemmas along with
Theorem 4.3 and Proposition 4.5.

###  5.1. The monomial groups $G(r,r,n)$

In this section we consider the restrictions of the reflection arrangements of
the monomial groups $W=G(r,r,n)$ for $r,n\geq 3$. Recall the intermediate
arrangements ${\mathscr{A}}_{\ell}^{k}(r)$ from §4.1.

For ${\mathscr{A}}={\mathscr{A}}_{3}^{k}(r)$ and $r\geq 3$,
${\mathscr{A}}^{\prime\prime}$ is of rank $2$, so
${\mathscr{A}}^{\prime\prime}$ and $({\mathscr{A}}^{\prime\prime},\kappa)$ are
inductively free, by Remark 2.11. So we may assume from now on that $\ell\geq
4$.

It follows from Proposition 4.5(ii) that if we restrict
${\mathscr{A}}^{k}_{\ell}(r)$ to a coordinate hyperplane, then we obtain
${\mathscr{A}}^{\ell-1}_{\ell-1}(r)={\mathscr{A}}(G(r,1,\ell-1))$. We consider
that case first, cf. [HR22, Ex. 1.14].

######  Corollary 5.1.

Let ${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ for $r\geq 3$, $\ell\geq 4$
and $1\leq k\leq\ell-1$. Fix a coordinate hyperplane $H_{0}$ in
${\mathscr{A}}$. Then $({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively
free.

###### Proof.

It follows from Proposition 4.5 that
${\mathscr{A}}^{\prime}\cong{\mathscr{A}}^{k-1}_{\ell}(r)$ is free. Moreover,
thanks to Theorem 4.2,
$\left({\mathscr{A}}_{\ell}^{k}(r)\right)^{H_{0}}\cong{\mathscr{A}}^{\ell-1}_{\ell-1}(r)={\mathscr{A}}(G(r,1,\ell-1))$
is inductively free. So the assertion follows from Theorem 1.4. ∎

Next we consider the case when $H_{0}$ is not a coordinate hyperplane in
${\mathscr{A}}$. Here we shall see that
$({\mathscr{A}}^{\prime\prime},\kappa)$ is not always inductively free, in
contrast to Corollary 5.1. We begin with the instance when
${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ itself is still inductively free.
In these instances $({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free
which is immediate by Theorem 1.3.

######  Lemma 5.2.

Let ${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ for $r\geq 3$, $\ell\geq 4$
and $k\in\\{\ell-2,\ell-1\\}$. Let $H_{0}=H_{i,j}(\zeta)$. Then
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free.

Next we consider the instances when
${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ is not inductively free, i.e.,
when $1\leq k\leq\ell-3$, cf. Theorem 4.3(i)(b) (cf. [HR22, Ex. 1.14]).

######  Lemma 5.3.

Let ${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ for $r\geq 3$, $\ell\geq 5$.
Fix $H_{0}=H_{s,t}(1)=\ker(x_{s}-x_{t})$. In the following instances
$({\mathscr{A}}^{\prime\prime},\kappa)$ is not inductively free:

  * (i)

$1\leq k\leq\ell-3$ and $1\leq s<t\leq\ell-3$;

  * (ii)

$1\leq k\leq\ell-4$ and $1\leq s\leq k<t$;

  * (iii)

$1\leq k\leq\ell-5$ and $1\leq k<s<t$.

###### Proof.

Set $Y_{i,j}(\zeta):=H_{0}\cap H_{i,j}(\zeta)$ and $Y_{i}:=H_{0}\cap H_{i}$ in
${\mathscr{A}}^{\prime\prime}$. Then one readily checks that

(5.4) $\kappa(Y)=\begin{cases}1&\text{ for }Y=Y_{i,j}(\zeta)\text{ and
}\\{s,t\\}\cap\\{i,j\\}=\varnothing,\\\ 1&\text{ for }Y=Y_{i}\text{ and
}i\notin\\{s,t\\}.\\\ \end{cases}$

First suppose we are in case (i). Define

$Z:=\bigcap_{\ell-2\leq i<j\leq\ell}{H_{i,j}(\zeta)}$

which is of rank $3$ in $L({\mathscr{A}})$ and let

$X:=H_{0}\cap Z=H_{0}\cap\left(\bigcap_{\ell-2\leq
i<j\leq\ell}{H_{i,j}(\zeta)}\right),$

which is of rank $3$ in $L({\mathscr{A}}^{\prime\prime})$. According to (5.4),
the multiplicity $\kappa_{X}$ of the localization
$\left(({\mathscr{A}}^{\prime\prime})_{X},\kappa_{X}\right)$ satisfies
$\kappa_{X}\equiv\mathbbm{1}$. Thus, it follows from the construction and the
hypotheses in (i) that the localization
$(({\mathscr{A}}^{\prime\prime})_{X},\kappa_{X})$ is isomorphic to the simple
reflection arrangement ${\mathscr{A}}(G(r,r,3))$. Owing to Theorem 4.1(i), the
latter is not inductively free. Therefore,
$({\mathscr{A}}^{\prime\prime},\kappa)$ is not inductively free, by Theorem
2.12.

Now suppose we are in case (ii) or (iii). Define

$Z:=\bigcap\limits_{\begin{subarray}{c}1\leq i<j\leq\ell\\\
\\{s,t\\}\cap\\{i,j\\}=\varnothing\end{subarray}}{H_{i,j}(\zeta)},$

which is of rank $\ell-2$ in $L({\mathscr{A}})$ and let

$X:=H_{0}\cap Z,$

which is of rank $\ell-2$ in $L({\mathscr{A}}^{\prime\prime})$. According to
(5.4), the multiplicity $\kappa_{X}$ of the localization
$\left(({\mathscr{A}}^{\prime\prime})_{X},\kappa_{X}\right)$ satisfies
$\kappa_{X}\equiv\mathbbm{1}$. Thus, it follows from the construction and the
hypotheses in (ii), respectively (iii), that the localization
$(({\mathscr{A}}^{\prime\prime})_{X},\kappa_{X})$ is isomorphic to the simple
reflection arrangement

(5.5)
$({\mathscr{A}}^{\prime\prime})_{X}=\begin{cases}{\mathscr{A}}^{k-1}_{\ell-2}&\text{
for }1\leq k\leq\ell-4\text{ and }1\leq s\leq k<t;\\\
{\mathscr{A}}^{k}_{\ell-2}&\text{ for }1\leq k\leq\ell-5\text{ and }1\leq
k<s<t.\\\ \end{cases}$

It follows from (5.5) and Theorem 4.3(i)(b) that
$({\mathscr{A}}^{\prime\prime})_{X}$ is not inductively free. Therefore,
$({\mathscr{A}}^{\prime\prime},\kappa)$ is not inductively free either, by
Theorem 2.12. ∎

So we are left to consider one instance when $\ell=4$.

######  Lemma 5.6.

Let ${\mathscr{A}}={\mathscr{A}}_{4}^{1}(r)$ for $r\geq 3$. Fix
$H_{0}=H_{i,j}(\zeta)$. Then $({\mathscr{A}}^{\prime\prime},\kappa)$ is
inductively free.

###### Proof.

We utilize Lemma 2.8. For that purpose set
${\mathscr{A}}_{0}:={\mathscr{A}}_{4}^{0}(r)$ and
${\mathscr{A}}_{1}:={\mathscr{A}}_{0}\cup\\{H_{1}=\ker
x_{1}\\}={\mathscr{A}}$. It follows from Theorems 1.5(ii), 4.1(i) and 1.3 that
$({\mathscr{A}}_{0}^{H_{0}},\kappa_{0})=({\mathscr{A}}_{4}^{0}(r)^{\prime\prime},\kappa)=({\mathscr{A}}(G(r,r,4))^{\prime\prime},\kappa)$
is inductively free with exponents $\\{r+1,2r+1,3r-3\\}$ (cf. Proposition
4.5(i)). By Lemma 2.8 and Proposition 4.5(ii), we observe that
$\left(({\mathscr{A}}_{1}^{H_{0}})^{H_{0}\cap
H_{1}},\kappa_{1}^{*}\right)=\left(({\mathscr{A}}_{1}^{H_{1}})^{H_{0}\cap
H_{1}},\kappa\right)=\left({\mathscr{A}}(G(r,1,3))^{\prime\prime},\kappa\right)$
which is inductively free, as it is of rank $2$, with exponents
$\\{r+1,2r+1\\}$. It follows from Definition 2.10 that also
$({\mathscr{A}}_{1}^{H_{0}},\kappa_{1})=({\mathscr{A}}_{4}^{1}(r)^{\prime\prime},\kappa)$
is inductively free. ∎

In view of Lemmas 5.2 and 5.3, we still have to consider the following
instances.

######  Lemma 5.7.

Let ${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ for $r\geq 3$, $\ell\geq 5$.
Fix $H_{0}=H_{s,t}(1)=\ker(x_{s}-x_{t})$. Let

  * (i)

$k\in\\{\ell-4,\ell-3\\}$ and $k<s<t$; or

  * (ii)

$k=\ell-3$ and $s\leq\ell-3<t$.

Then $({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free.

###### Proof.

It follows from Proposition 4.5(ii) that
${\mathscr{A}}^{\prime\prime}={\mathscr{A}}_{\ell-1}^{k+1}(r)$ in (i), and
${\mathscr{A}}^{\prime\prime}={\mathscr{A}}_{\ell-1}^{\ell-3}(r)$ in (ii). In
any event, ${\mathscr{A}}^{\prime\prime}$ is inductively free, by Theorem
4.3(i)(b). In each case, we construct an induction table from the inductively
free arrangement $({\mathscr{A}}^{\prime\prime},\mathbbm{1})$ to
$({\mathscr{A}}^{\prime\prime},\kappa)$.

Let
${\mathscr{A}}_{0}={\mathscr{A}}^{\ell-4}_{\ell}(r),{\mathscr{A}}_{1}={\mathscr{A}}^{\ell-4}_{\ell}(r)\cup\\{H_{1}=\ker(x_{\ell-3})\\}={\mathscr{A}}^{\ell-3}_{\ell}(r)$,
then ${\mathscr{A}}_{0}$ and ${\mathscr{A}}_{1}$ are free, thanks to
Proposition 4.5(i). Now use Lemma 2.8(2) to see that for the triple
$({\mathscr{A}}^{H_{0}}_{0},\kappa_{0}),({\mathscr{A}}^{H_{0}}_{1},\kappa_{1}),\left(({\mathscr{A}}^{H_{0}}_{1})^{H_{0}\cap
H_{1}},\kappa_{1}^{*}\right)$ we get

$\left(({\mathscr{A}}^{H_{0}}_{1})^{H_{0}\cap
H_{1}},\kappa_{1}^{*}\right)=\left(({\mathscr{A}}^{H_{1}}_{1})^{H_{0}\cap
H_{1}},\kappa\right)=({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{\prime\prime},\kappa).$

The inclusion
$\exp({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{\prime\prime},\kappa)\subset\exp({\mathscr{A}}^{H_{0}}_{0},\kappa_{0})$
follows from Proposition 4.5(i) and Theorem 1.2. Now by Theorems 1.3 and
4.1(i) we see that
$({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{\prime\prime},\kappa)$ is inductively
free. So it is sufficient to show the inductive freeness of
$({\mathscr{A}}^{H_{0}}_{0},\kappa_{0})=({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\kappa_{0})$
to deduce part (i).

Using Proposition 4.5(ii), we get
$({\mathscr{A}}^{\ell-4}_{\ell}(r)^{H_{0}},\kappa)=({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\kappa)$
and the Ziegler multiplicity is given by (with the notation as in Lemma 5.3):

$\kappa(Y)=\begin{cases}r-1&\text{ for
}Y=Y_{s}:=\cap_{k=1}^{r}H_{s,t}(\zeta^{k}),\\\ 2&\text{ for
}Y=Y_{i,t}(\zeta),s\neq i<t,\\\ 2&\text{ for }Y=Y_{s,i}(\zeta),t<i,\\\
1&\text{ for }Y=Y_{i,j}(\zeta)\text{ and
}\\{s,t\\}\cap\\{i,j\\}=\varnothing,\\\ 1&\text{ for }Y=Y_{i},i\neq
s.\end{cases}$

Thanks to Theorem 4.3(i)(b),
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\mathbbm{1})$ is inductively free. So
starting at the simple multiplicity $\mathbbm{1}$ we have to add $r-2$
hyperplanes of type $Y_{s}$ and $(\ell-2)r$ hyperplanes which are of type
$Y_{s,j}(\zeta),(t<j)$ or $Y_{i,t}(\zeta),(i<t)$. We define multiplicities
$\mu_{i}$ on ${\mathscr{A}}^{\ell-3}_{\ell-1}(r)$ so that
$\mu_{0}=\mathbbm{1},|\mu_{i+1}|-|\mu_{i}|=1$ and
$\mathbbm{1}=\mu_{0}\leq\dots\leq\mu_{(\ell-1)r-2}=\kappa$ as follows: Start
by adding all the hyperplanes of type $Y_{1,t}(\zeta)$, then all of type
$Y_{2,t}(\zeta)$ up to type $Y_{t-1,t}(\zeta)$. Continue by adding all of type
$Y_{s,t+1}(\zeta)$ up to type $Y_{s,\ell}(\zeta)$. At the end add all the
hyperplanes of type $Y_{s}$. We present the induction in Table 2 below.

We now show that the $({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\mu_{i})$ provide a
chain of inductively free arrangements from
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\mathbbm{1})$ to
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\kappa)$, proving part (i) of the lemma.

While adding the hyperplanes of type $Y_{s,j}(\zeta)$ and $Y_{i,t}(\zeta)$ the
Euler multiplicity coincides with the Ziegler multiplicity, giving
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\mu_{i}^{*})=({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
in every step for $1\leq i\leq(\ell-2)r$ which is inductively free, because of
Theorems 1.3 and 4.3(i)(b). This is not affected by the different types of
occurring restrictions (namely ${\mathscr{A}}^{\ell-4}_{\ell-2}(r)$ and
${\mathscr{A}}^{\ell-3}_{\ell-2}(r)$).

To show the equality between the multiplicities, let $H=Y_{i,t}(\zeta)$ be
arbitrary and let $\mu_{p}$ be the unique multiplicity so that $\mu_{p}(H)=2$
and $\mu_{p-1}(H)=1$. There are three different types of localizations for
$X\in{\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{H}$ in
${\mathscr{A}}^{\ell-3}_{\ell-1}(r)$. The first kind has got $2$, the second
kind $3$, and the third kind has got $r+2$ or $r+3$ elements, depending on the
indices of the hyperplanes. The three types are as follows:

  * (a)

${\mathscr{A}}^{\ell-3}_{\ell-1}(r)_{X}=\\{Y_{i,t}(\zeta),Y_{a,b}(\zeta^{\prime})\\}$
or $\\{Y_{i,t}(\zeta),Y_{a}\\}$ with $a,b\not\in\\{i,t\\}$,

  * (b)

${\mathscr{A}}^{\ell-3}_{\ell-1}(r)_{X}=\\{Y_{i,t}(\zeta),Y_{i,a}(\zeta^{\prime}),Y_{a,t}(\zeta^{\prime\prime})\\}$
or
$\\{Y_{i,t}(\zeta),Y_{i,j}(\zeta^{\prime}),Y_{s,j}(\zeta^{\prime\prime})\\}$,

  * (c)

${\mathscr{A}}^{\ell-3}_{\ell-1}(r)_{X}=$
$\\{Y_{i},Y_{s},Y_{i,t}(1),Y_{i,t}(2),\dots,Y_{i,t}(r)\\}$, respectively
${\mathscr{A}}^{\ell-3}_{\ell-1}(r)_{X}=\\{Y_{s},Y_{s,j}(1),Y_{s,j}(2),\dots,Y_{s,j}(r)\\}$.

In case of (a), using Proposition 2.6(1) gives $\mu_{p}^{*}(X)=1=\kappa(X)$,
since $(\mu_{p})_{X}(Y_{a,b}(\zeta^{\prime}))=(\mu_{p})_{X}(Y_{a})=1$. The
cases (b) and (c) are handled similarly, using Proposition 2.6(2). This gives
the desired equality
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\mu_{i}^{*})=({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
at each step for $1\leq i\leq(\ell-2)r$. Note that because of Theorems 1.3 and
4.3(i)(b) every restriction
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$ is inductively
free. Now use Theorem 1.2 to see

$\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\mu_{i}^{*})=\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)=\\{r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\},$

which gives
$\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\mu_{(\ell-2)r})=\\{r+1,2r+1,\dots,$
$(\ell-2)r-1,(\ell-2)r+1\\}$, by using Theorem 2.9 and this shows that
$({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\mu_{(\ell-2)r})$ is inductively free.

Now $r-2$ hyperplanes of type $Y_{s}$ are left to be added. Owing to Theorem
1.5, for an arbitrary coordinate hyperplane
$H_{j}\in{\mathscr{A}}^{\ell-1}_{\ell-1}(r)$, the restriction
$\left({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{H_{j}},\kappa\right)$ is
inductively free with exponents
$\exp\left({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{H_{j}},\kappa\right)=\\{r+1,2r+1,\dots,$
$(\ell-2)r+1\\}$ and $({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\kappa)$ is free
with exponents
$\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\kappa)=\\{r+1,2r+1,\dots,(\ell-2)r+1,(\ell-1)r-3\\}$.
So it is sufficient to show that
$\left({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{Y_{s}},\mu_{i}^{*}\right)=\left({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{H_{j}},\kappa\right)$
for every $i$ with $(\ell-2)r+1\leq i\leq(\ell-1)r-2$.
Showing that
$\left({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{Y_{s}},\mu_{(\ell-2)r+1}^{*}\right)=\left({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{H_{j}},\kappa\right)$
is once again straightforward, relying on Proposition 2.6. Thanks to
Definition 2.4 and Remark 2.5, it is sufficient to show that all localizations
$\left({\mathscr{A}}^{\ell-3}_{\ell-1}(r)_{X},(\mu_{i})_{X}\right)$ for any
$X\in{\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{Y_{s}}$ and $(\ell-2)r+1\leq
i\leq(\ell-1)r-2$ share a common exponent, which implies in turn that the
Euler multiplicity does not change. This is obvious for the localizations with
two elements, since Proposition 2.6(1) does not rely on the multiplicity of
$Y_{s}$. The remaining localizations are
$X_{1}=\\{Y_{s},Y_{a},Y_{a,t}(1),\dots,Y_{a,t}(r)\\}$ and
$X_{2}=\\{Y_{s},Y_{s,j}(1),\dots,Y_{s,j}(r)\\},(j\neq t)$. We have shown above
that $\mu_{(\ell-2)r+1}^{*}(X_{1})=r+1$ which implies that
$\exp\left(X_{1},\mu_{(\ell-2)r+1}|_{X_{1}}\right)=\\{r+1,r+2\\}$ since
$|\mu_{(\ell-2)r+1}|_{X_{1}}|=2r+3$. We also have
$\left({\mathscr{A}}_{X_{1}},\mu_{(\ell-1)r-2}|_{X_{1}}\right)=\left({\mathscr{A}}_{X_{1}},\kappa|_{X_{1}}\right)\simeq\left({\mathscr{A}}^{1}_{3}(r)^{H_{2,3}(\zeta)},\kappa\right)$,
as $2$-arrangements which shows that
$\exp({\mathscr{A}}_{X_{1}},\mu_{(\ell-1)r-2}|_{X_{1}})=\\{r+1,2r-1\\}$ and we
see that $r+1$ is the common exponent. In particular, we get
$\mu_{i}^{*}(X_{1})=r+1$ for an arbitrary $(\ell-2)r+1\leq i\leq(\ell-1)r-2$.
The argument for $X_{2}$ is identical with the only difference that
$\left({\mathscr{A}}_{X_{2}},\mu_{(\ell-1)r-2}|_{X_{2}}\right)\simeq\left({\mathscr{A}}^{0}_{3}(r)^{H_{1,2}(\zeta)},\kappa\right)$
and the exponents change from $\\{r+1,r+1\\}$ to $\\{r+1,2r-2\\}$. This
completes the proof.

$\exp({\mathscr{A}}^{\prime},\mu^{\prime})$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\mathbbm{1})=$ | $Y_{1,t}(1)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
$\\{1,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ |  |
$\\{2,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ | $Y_{1,t}(2)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
⋮ | ⋮ | ⋮
$\\{r,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ | $Y_{1,t}(r)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
$\\{r+1,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ | $Y_{2,t}(1)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
⋮ | ⋮ | ⋮
$\\{(t-1)r-1,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ | $Y_{t-1,t}(r)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
$\\{(t-1)r,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ | $Y_{s,t+1}(1)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
⋮ | ⋮ | ⋮
$\\{(\ell-2)r,r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1\\}$ | $Y_{s,\ell}(r)$ | $\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r)^{\prime\prime},\kappa)$
$\\{r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r-1,(\ell-2)r+1\\}$ | $Y_{s}$ | $\exp({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{\prime\prime},\kappa)$
⋮ | ⋮ | ⋮
$\\{r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r+1,(\ell-1)r-4\\}$ | $Y_{s}$ | $\exp({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{\prime\prime},\kappa)$
$\exp({\mathscr{A}}^{\ell-3}_{\ell-1}(r),\kappa)=$ |  |
$\\{r+1,2r+1,\dots,(\ell-3)r+1,(\ell-2)r+1,(\ell-1)r-3\\}$ |  |
Table 2. Induction table for
${\mathscr{A}}={\mathscr{A}}^{l-4}_{l}(r),({\mathscr{A}}^{H},\kappa)=({\mathscr{A}}^{l-3}_{l-1}(r),\kappa)$

To show (ii) let ${\mathscr{A}}={\mathscr{A}}^{\ell-3}_{\ell}(r)$ and
$H_{0}=H_{\ell-3,t}(1)$ with $\ell-3<t\leq\ell$ (we get $s=\ell-3$ after a
permutation of basis vectors). Let
${\mathscr{A}}_{0}={\mathscr{A}}\backslash\\{H_{1}=\ker(x_{\ell-3})\\}$ and
${\mathscr{A}}_{1}={\mathscr{A}}$, then
${\mathscr{A}}_{0}\simeq{\mathscr{A}}^{\ell-4}_{\ell}(r)$ and we are in the
setting of (i) for $k=\ell-4$. Now use Lemma 2.8 as in the proof of (i) to see
that
$({\mathscr{A}}^{\prime\prime},\kappa)=({\mathscr{A}}_{1}^{H_{0}},\kappa_{1})$
is inductively free because $({\mathscr{A}}_{0}^{H_{0}},\kappa_{0})$ and
$\left(({\mathscr{A}}^{H_{0}}_{1})^{H_{0}\cap
H_{1}},\kappa_{1}^{*}\right)=\left(({\mathscr{A}}^{H_{1}}_{1})^{H_{0}\cap
H_{1}},\kappa\right)=({\mathscr{A}}^{\ell-1}_{\ell-1}(r)^{\prime\prime},\kappa)$
are. ∎

######  Remark 5.8.

Owing to Corollary 5.1 and Lemma 5.3, for
${\mathscr{A}}={\mathscr{A}}_{\ell}^{k}(r)$ with $r\geq 3$, $\ell\geq 5$ and
$1\leq k\leq\ell-3$, it depends on the choice of $H_{0}$ whether
$({\mathscr{A}}^{\prime\prime},\kappa)$ is inductively free or not.

###  5.2. Groups of type $G_{33}$ and $G_{34}$

Here we consider the outstanding restriction of the reflection arrangements of
$G_{33}$ and $G_{34}$ of dimension at least $4$.

Since the pointwise stabilizer $W_{X}$ of $X$ in $L({\mathscr{A}}(W))$ is
itself a complex reflection group, by [S64, Thm. 1.5], following [OS82, §3,
App.] (cf. [OT92, §6.4, App. C]), we may label the $W$-orbit of $X\in
L({\mathscr{A}}(W))$ by the type $T$ say, of $W_{X}$; so we usually denote
such a restriction ${\mathscr{A}}(W)^{X}$ simply by the pair $(W,T)$ in Tables
3 and 4.

It follows from the deletion part of Theorem 2.1 that if both ${\mathscr{A}}$
and ${\mathscr{A}}^{\prime\prime}$ are free and
$\exp{\mathscr{A}}^{\prime\prime}\subset\exp{\mathscr{A}}$, then also
${\mathscr{A}}^{\prime}$ is free. If in this case in addition
${\mathscr{A}}^{\prime\prime}$ is inductively free, then so is
$({\mathscr{A}}^{\prime\prime},\kappa)$, thanks to Theorem 1.4. Thus the
following is a consequence of Theorems 1.4, 2.1, and 4.3(i)(c), the fact that
all restrictions of reflection arrangements are free, and the data in [OS82,
Tables 10, 11] (cf. [OT92, C.14 - C.17]).

######  Lemma 5.9.

Let $W$ be of type $G_{33}$ or $G_{34}$ with reflection arrangement
${\mathscr{A}}(W)$. Let ${\mathscr{A}}={\mathscr{A}}(W)^{X}$ for $X\in
L({\mathscr{A}}(W))$ with restriction ${\mathscr{A}}^{H}$ as in Table 3. Then
$({\mathscr{A}}^{H},\kappa)$ is inductively free for any $H\in{\mathscr{A}}$.

$W$ | ${\mathscr{A}}={\mathscr{A}}(W)^{X}$ | ${\mathscr{A}}^{H}$ | $\operatorname{rank}{\mathscr{A}}^{H}$
---|---|---|---
$G_{33}$ | $(G_{33},A_{1})$ | $(G_{33},A_{1}^{2})$ | $3$
$G_{34}$ | $(G_{34},A_{1}^{2})$ | $(G_{34},A_{1}^{3})$ | $3$
| $(G_{34},A_{2})$ | $(G_{34},A_{1}A_{2})$ | $3$
Table 3. Restrictions of $G_{33}$ and $G_{34}$, I

The remaining instances of restrictions of reflection arrangements of type
$G_{33}$ or $G_{34}$ to be considered are listed in Table 4. Theorem 1.4 does
not apply here, as ${\mathscr{A}}^{\prime}$ is not free. However, note that in
each case when ${\mathscr{A}}^{H}$ is of rank $3$, ${\mathscr{A}}^{H}$ is
inductively free, thanks to Theorem 4.3(i)(c). In each of these instances we
provide a suitable induction table from the simple arrangement
${\mathscr{A}}^{H}$ to the Ziegler multiplicity.

######  Lemma 5.10.

Let $W$ be of type $G_{33}$ or $G_{34}$ with reflection arrangement
${\mathscr{A}}(W)$. Let ${\mathscr{A}}={\mathscr{A}}(W)^{X}$ for $X\in
L({\mathscr{A}}(W))$ with restriction ${\mathscr{A}}^{H}$ as in Table 4. Then
$({\mathscr{A}}^{H},\kappa)$ is inductively free if and only if $\dim X=4$.

$W$ | ${\mathscr{A}}={\mathscr{A}}(W)^{X}$ | ${\mathscr{A}}^{H}$ | $\operatorname{rank}{\mathscr{A}}^{H}$
---|---|---|---
$G_{33}$ | $(G_{33},A_{1})$ | $(G_{33},A_{2})$ | $3$
$G_{34}$ | $(G_{34},A_{1})$ |  $(G_{34},A_{1}^{2})$, $(G_{34},A_{2})$ | $4$
| $(G_{34},A_{1}^{2})$ |  $(G_{34},A_{1}A_{2})$, $(G_{34},A_{3})$ | $3$
| $(G_{34},A_{2})$ |  $(G_{34},A_{3})$, $(G_{34},G(3,3,3))$ | $3$
Table 4. Restrictions of $G_{33}$ and $G_{34}$, II

###### Proof.

Firstly, in each instance in Table 4 when ${\mathscr{A}}$ is of rank $4$, we
present an induction table for $({\mathscr{A}}^{\prime\prime},\kappa)$; see
Tables 5, 6, and 7 below.

Let $\zeta=e^{\frac{2\pi i}{3}}$ be a primitive third root of unity. We fix
the order of the linear forms and refer to the corresponding hyperplane by its
position in the defining polynomial of ${\mathscr{A}}^{H}$ in each case.

The defining polynomial of $(G_{33},A_{1})$ is given by

$\displaystyle Q(G_{33},A_{1})=\ $
$\displaystyle(x_{1}-x_{4})(x_{3}-x_{4})(x_{3}+(-\zeta^{2})x_{4})(2x_{1}-x_{2}+x_{3}+x_{4})(x_{1}-x_{3})(x_{1}-\zeta
x_{3})(x_{3}-\zeta x_{4})(x_{1}-\zeta x_{4})$
$\displaystyle(2x_{1}-x_{2}+\zeta
x_{3}+\zeta^{2}x_{4})(x_{1}-\zeta^{2}x_{4})(x_{1}-\zeta^{2}x_{3})(-\zeta
x_{1}-x_{2}+\zeta x_{3}+x_{4})(2x_{1}-x_{2}+\zeta^{2}x_{3}+\zeta x_{4})$
$\displaystyle(-\zeta x_{1}-x_{2}+x_{3}+\zeta
x_{4})(-\zeta^{2}x_{1}-x_{2}+x_{3}+\zeta^{2}x_{4})(-\zeta^{2}x_{1}-x_{2}+\zeta^{2}x_{3}+x_{4})((-\zeta-2\zeta^{2})x_{1})$
$\displaystyle(-\zeta^{2}x_{1}-\zeta^{2}x_{2}+\zeta^{2}x_{3}+\zeta^{2}x_{4})(-\zeta
x_{1}-x_{2}+\zeta^{2}x_{3}+\zeta^{2}x_{4})(-\zeta^{2}x_{1}-x_{2}+\zeta
x_{3}+\zeta x_{4})$ $\displaystyle(-\zeta^{2}x_{1}-\zeta^{2}x_{2}+x_{3}+\zeta
x_{4})(2x_{1}-\zeta x_{2}+x_{3}+\zeta
x_{4})(2x_{1}-\zeta^{2}x_{2}+x_{3}+\zeta^{2}x_{4})(2x_{1}-\zeta
x_{2}+\zeta^{2}x_{3}+\zeta^{2}x_{4})$
$\displaystyle(-\zeta^{2}x_{1}-\zeta^{2}x_{2}+\zeta x_{3}+x_{4})(2x_{1}-\zeta
x_{2}+\zeta
x_{3}+x_{4})(2x_{1}-\zeta^{2}x_{2}+\zeta^{2}x_{3}+x_{4})(2x_{1}-\zeta^{2}x_{2}+\zeta
x_{3}+\zeta x_{4}).$

The Ziegler restriction of $(G_{33},A_{1})$ with respect to
$H=\ker(x_{1}-x_{2})$ gives $((G_{33},A_{2}),\kappa)$ with defining polynomial

$\displaystyle Q((G_{33},A_{2}),\kappa)=\ $
$\displaystyle(x_{2}-x_{3})^{2}(x_{2}-\zeta^{2}x_{3})^{2}(-x_{1}+x_{2}+3x_{3})(x_{2}-\zeta
x_{3})^{2}((-2\zeta-\zeta^{2})x_{3})^{3}(-x_{1}+\zeta
x_{2}+(-2\zeta-\zeta^{2})x_{3})^{2}$
$\displaystyle(-x_{1}+\zeta^{2}x_{2}+(-\zeta-2\zeta^{2})x_{3})^{2}(-x_{1}+x_{2})^{3}(-x_{1}+\zeta^{2}x_{2}+(-\zeta+\zeta^{2})x_{3})^{2}(-x_{1}+\zeta
x_{2}+(\zeta-\zeta^{2})x_{3})^{2}$
$\displaystyle((-\zeta^{2})x_{1}+x_{2}+(\zeta-\zeta^{2})x_{3})^{2}((-\zeta^{2})x_{1}+\zeta
x_{2}+(-\zeta-2\zeta^{2})x_{3})^{2}((-\zeta)x_{1}+\zeta
x_{2}+3x_{3})((-\zeta^{2})x_{1}+\zeta^{2}x_{2}+3x_{3}).$

$\exp({\mathscr{A}}^{\prime},\mu^{\prime})$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp((G_{33},A_{2}),\mathbbm{1})=\\{1,6,7\\}$ | $\alpha_{5}$ | $\\{6,7\\}$
$\\{2,6,7\\}$ | $\alpha_{5}$ | $\\{6,7\\}$
$\\{3,6,7\\}$ | $\alpha_{8}$ | $\\{6,7\\}$
$\\{4,6,7\\}$ | $\alpha_{8}$ | $\\{6,7\\}$
$\\{5,6,7\\}$ | $\alpha_{4}$ | $\\{6,7\\}$
$\\{6,6,7\\}$ | $\alpha_{11}$ | $\\{6,7\\}$
$\\{6,7,7\\}$ | $\alpha_{7}$ | $\\{6,7\\}$
$\\{6,7,8\\}$ | $\alpha_{12}$ | $\\{7,8\\}$
$\\{7,7,8\\}$ | $\alpha_{10}$ | $\\{7,8\\}$
$\\{7,8,8\\}$ | $\alpha_{9}$ | $\\{7,8\\}$
$\\{7,8,9\\}$ | $\alpha_{6}$ | $\\{7,9\\}$
$\\{7,9,9\\}$ | $\alpha_{2}$ | $\\{7,9\\}$
$\\{7,9,10\\}$ | $\alpha_{1}$ | $\\{7,9\\}$
$\exp((G_{33},A_{2}),\kappa)=\\{7,9,11\\}$ |  |
Table 5. Induction table for
${\mathscr{A}}=(G_{33},A_{1}),({\mathscr{A}}^{H},\kappa)=((G_{33},A_{2}),\kappa)$

The defining polynomial of $(G_{34},A_{1}^{2})$ is given by

$\displaystyle Q(G_{34},A_{1}^{2})=\ $
$\displaystyle(x_{1}-x_{4})((\zeta+2\zeta^{2})x_{4})(2x_{1}+\zeta^{2}x_{2}+\zeta
x_{3}-\zeta x_{4})(x_{2}-\zeta x_{3})(x_{1}-\zeta^{2}x_{4})(x_{1}-\zeta
x_{4})(2x_{1}+x_{2}+x_{3}-\zeta x_{4})$ $\displaystyle(-\zeta
x_{1}+\zeta^{2}x_{2}+\zeta x_{3}+2x_{4})(2x_{1}+\zeta^{2}x_{2}+\zeta
x_{3}+2\zeta x_{4})(-\zeta x_{1}+\zeta^{2}x_{2}+\zeta
x_{3}-x_{4})(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta x_{3}+2\zeta^{2}x_{4})$
$\displaystyle(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta
x_{3}-\zeta^{2}x_{4})(-\zeta
x_{1}+x_{2}+x_{3}+2x_{4})(2x_{1}+x_{2}+x_{3}+2\zeta x_{4})(-\zeta
x_{1}+x_{2}+x_{3}-x_{4})$
$\displaystyle(-\zeta^{2}x_{1}+x_{2}+x3+2\zeta^{2}x_{4})(-\zeta^{2}x_{1}+x_{2}+x_{3}-\zeta^{2}x_{4})((-\zeta-2\zeta^{2})x_{1})(-\zeta^{2}x_{1}+\zeta
x_{2}+x_{3}-x_{4})$
$\displaystyle(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3}-x_{4})(x_{1}-\zeta
x_{2})(x_{1}-\zeta^{2}x_{3})(-\zeta
x_{1}+\zeta^{2}x_{2}+x_{3}+2\zeta^{2}x_{4})(2x_{1}+x_{2}+\zeta^{2}x_{3}-x_{4})$
$\displaystyle(2x_{1}+\zeta x_{2}+x_{3}+2\zeta^{2}x_{4})(2x_{1}+\zeta
x_{2}+\zeta
x_{3}-x_{4})(2x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3}+2\zeta^{2}x_{4})(x_{2}-\zeta^{2}x_{4})(x_{3}-\zeta
x_{4})$ $\displaystyle(2x_{1}+x_{2}+\zeta
x_{3}-\zeta^{2}x_{4})(-\zeta^{2}x_{1}+\zeta
x_{2}+x_{3}+2x_{4})(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3}+2x_{4})(-\zeta
x_{1}+\zeta^{2}x_{2}+x_{3}-\zeta^{2}x_{4})$
$\displaystyle(2x_{1}+x_{2}+\zeta^{2}x_{3}+2x_{4})(2x_{1}+\zeta
x_{2}+x_{3}-\zeta^{2}x_{4})(2x_{1}+\zeta x_{2}+\zeta
x_{3}+2x_{4})(2x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3}-\zeta^{2}x_{4})(x_{2}-\zeta
x_{4})$
$\displaystyle(x_{3}-x_{4})(2x_{1}+\zeta^{2}x_{2}+x_{3}-x_{4})(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+x_{3}-\zeta
x_{4})(-\zeta x_{2}+x_{2}+\zeta x_{3}+2\zeta
x_{4})(x_{1}-\zeta^{2}x_{2})(x_{1}-x_{3})$
$\displaystyle(x_{2}-x_{4})(x_{3}-\zeta^{2}x_{4})(2x_{1}+x_{2}+\zeta
x_{3}+2\zeta^{2}x_{4})(2x_{1}+\zeta^{2}x_{2}+x_{3}+2x_{4})(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+x_{3}+2\zeta
x_{4})$ $\displaystyle(-\zeta x_{1}+x_{2}+\zeta x_{3}-\zeta
x_{4})(x_{1}-x_{2})(x_{1}-\zeta x_{3})(2x_{1}+\zeta x_{2}+\zeta^{2}x_{3}-\zeta
x_{4})(2x_{1}+\zeta x_{2}+\zeta^{2}x_{3}+2\zeta x_{4})(x_{2}-x_{3})$
$\displaystyle(x_{2}-\zeta^{2}x_{3}).$

The Ziegler restriction of $(G_{34},A_{1}^{2})$ with respect to
$H=\ker((\zeta+2\zeta^{2})x_{4})$ gives $((G_{34},A_{1}A_{2}),\kappa)$ with
defining polynomial

$\displaystyle Q((G_{34},A_{1}A_{2}),\kappa)=\ $
$\displaystyle(x_{1})^{4}(2x_{1}+\zeta^{2}x_{2}+\zeta x_{3})^{2}(x_{2}-\zeta
x_{3})(2x_{1}+x_{2}+x_{3})^{2}(-\zeta x_{1}+\zeta^{2}x_{2}+\zeta
x_{3})^{2}(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta x_{3})^{2}$
$\displaystyle(-\zeta
x_{1}+x_{2}+x_{3})^{2}(-\zeta^{2}x_{1}+x_{2}+x_{3})^{2}(-\zeta^{2}x_{1}+\zeta
x_{2}+x_{3})^{2}(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3})^{2}(x_{1}-\zeta
x_{2})(x_{1}-\zeta^{2}x_{3})$ $\displaystyle(-\zeta
x_{1}+\zeta^{2}x_{2}+x_{3})^{2}(2x_{1}+x_{2}+\zeta^{2}x_{3})^{2}(2x_{1}+\zeta
x_{2}+x_{3})^{2}(2x_{1}+\zeta x_{2}+\zeta
x_{3})^{2}(2x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3})^{2}(x_{2})^{3}$
$\displaystyle(x_{3})^{3}(2x_{1}+x_{2}+\zeta
x_{3})^{2}(2x_{1}+\zeta^{2}x_{2}+x_{3})^{2}(-\zeta^{2}x_{1}+\zeta^{2}x_{2}+x_{3})^{2}(-\zeta
x_{1}+x_{2}+\zeta x_{3})^{2}(x_{1}-\zeta^{2}x_{2})(x_{1}-x_{3})$
$\displaystyle(x_{1}-x_{2})(x_{1}-\zeta x_{3})(2x_{1}+\zeta
x_{2}+\zeta^{2}x_{3})^{2}(x_{2}-x_{3})(x_{2}-\zeta^{2}x_{3}).$

The Ziegler restriction of $(G_{34},A_{1}^{2})$ with respect to
$H=\ker(x_{1}-x_{4})$ gives $((G_{34},A_{3}),\kappa)$ with defining polynomial

$\displaystyle Q((G_{34},A_{3}),\kappa)=\ $
$\displaystyle((\zeta+2\zeta^{2})x_{3})^{4}(\zeta^{2}x_{1}+\zeta
x_{2}+(-3\zeta-2\zeta^{2})x_{3})^{2}(x_{1}-\zeta
x_{2})(x_{1}+x_{2}+(-3\zeta-2\zeta^{2})x_{3})^{2}$
$\displaystyle(\zeta^{2}x_{1}+\zeta
x_{2}-2\zeta^{2}x_{3})^{3}(\zeta^{2}x_{1}+\zeta
x_{2}+\zeta^{2}x_{3})^{4}(x_{1}+x_{2}-2\zeta^{2}x_{3})^{3}(x_{1}+x_{2}+\zeta^{2}x_{3})^{4}(-\zeta
x_{1}+x_{3})^{2}(-\zeta^{2}x_{1}+x_{3})^{2}$
$\displaystyle(\zeta^{2}x_{1}+x_{2}+(-\zeta+2\zeta^{2})x_{3})^{2}(\zeta
x_{1}+x_{2}+(-2\zeta-3\zeta^{2})x_{3})^{2}(\zeta^{2}x_{1}+\zeta^{2}x_{2}+(-2\zeta-3\zeta^{2})x_{3})^{2}(\zeta^{2}x_{1}+x_{2}+x_{3})^{4}$
$\displaystyle(x_{1}+\zeta^{2}x_{2}+4x_{3})(\zeta x_{1}+\zeta
x_{2}+4x_{3})(x_{1}-\zeta
x_{3})^{2}(x_{2}-x_{3})^{2}(x_{1}-x_{3})^{2}(x_{2}-\zeta^{2}x_{3})^{2}(x_{1}+\zeta
x_{2}-2\zeta x_{3})^{3}$
$\displaystyle(\zeta^{2}x_{1}+x_{2}+4x_{3})(\zeta^{2}x_{1}+x_{2}+(2\zeta-\zeta^{2})x_{3})^{2}(x_{1}-x_{2})(x_{1}-\zeta^{2}x_{2}).$

$\exp({\mathscr{A}}^{\prime},\mu^{\prime})$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp((G_{34},A_{1}A_{2}),\mathbbm{1})=\\{1,13,16\\}$ | $\alpha_{18}$ | $\\{13,16\\}$
$\\{2,13,16\\}$ | $\alpha_{18}$ | $\\{13,16\\}$
$\\{3,13,16\\}$ | $\alpha_{22}$ | $\\{13,16\\}$
$\\{4,13,16\\}$ | $\alpha_{9}$ | $\\{13,16\\}$
$\\{5,13,16\\}$ | $\alpha_{8}$ | $\\{13,16\\}$
$\\{6,13,16\\}$ | $\alpha_{7}$ | $\\{13,16\\}$
$\\{7,13,16\\}$ | $\alpha_{13}$ | $\\{13,16\\}$
$\\{8,13,16\\}$ | $\alpha_{6}$ | $\\{13,16\\}$
$\\{9,13,16\\}$ | $\alpha_{10}$ | $\\{13,16\\}$
$\\{10,13,16\\}$ | $\alpha_{23}$ | $\\{13,16\\}$
$\\{11,13,16\\}$ | $\alpha_{5}$ | $\\{13,16\\}$
$\\{12,13,16\\}$ | $\alpha_{19}$ | $\\{13,16\\}$
$\\{13,13,16\\}$ | $\alpha_{19}$ | $\\{13,16\\}$
$\\{13,14,16\\}$ | $\alpha_{1}$ | $\\{13,16\\}$
$\\{13,15,16\\}$ | $\alpha_{1}$ | $\\{13,16\\}$
$\\{13,16,16\\}$ | $\alpha_{1}$ | $\\{13,16\\}$
$\\{13,16,17\\}$ | $\alpha_{28}$ | $\\{13,17\\}$
$\\{13,17,17\\}$ | $\alpha_{21}$ | $\\{13,17\\}$
$\\{13,17,18\\}$ | $\alpha_{20}$ | $\\{13,17\\}$
$\\{13,17,19\\}$ | $\alpha_{16}$ | $\\{13,19\\}$
$\\{13,18,19\\}$ | $\alpha_{14}$ | $\\{13,19\\}$
$\\{13,19,19\\}$ | $\alpha_{17}$ | $\\{13,19\\}$
$\\{13,19,20\\}$ | $\alpha_{15}$ | $\\{13,19\\}$
$\\{13,19,21\\}$ | $\alpha_{4}$ | $\\{13,19\\}$
$\\{13,19,22\\}$ | $\alpha_{2}$ | $\\{13,19\\}$
$\exp((G_{34},A_{1}A_{2}),\kappa)=\\{13,19,23\\}$ |  |

$\exp({\mathscr{A}}^{\prime},\mu^{\prime})$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp((G_{34},A_{3}),\mathbbm{1})=\\{1,11,13\\}$ | $\alpha_{23}$ | $\\{11,13\\}$
$\\{2,11,13\\}$ | $\alpha_{20}$ | $\\{11,13\\}$
$\\{3,11,13\\}$ | $\alpha_{19}$ | $\\{11,13\\}$
$\\{4,11,13\\}$ | $\alpha_{18}$ | $\\{11,13\\}$
$\\{5,11,13\\}$ | $\alpha_{17}$ | $\\{11,13\\}$
$\\{6,11,13\\}$ | $\alpha_{13}$ | $\\{11,13\\}$
$\\{7,11,13\\}$ | $\alpha_{12}$ | $\\{11,13\\}$
$\\{8,11,13\\}$ | $\alpha_{11}$ | $\\{11,13\\}$
$\\{9,11,13\\}$ | $\alpha_{10}$ | $\\{11,13\\}$
$\\{10,11,13\\}$ | $\alpha_{9}$ | $\\{11,13\\}$
$\\{11,11,13\\}$ | $\alpha_{4}$ | $\\{11,13\\}$
$\\{11,12,13\\}$ | $\alpha_{2}$ | $\\{11,13\\}$
$\\{11,13,13\\}$ | $\alpha_{14}$ | $\\{13,13\\}$
$\\{12,13,13\\}$ | $\alpha_{8}$ | $\\{13,13\\}$
$\\{13,13,13\\}$ | $\alpha_{6}$ | $\\{13,13\\}$
$\\{13,13,14\\}$ | $\alpha_{1}$ | $\\{13,13\\}$
$\\{13,13,15\\}$ | $\alpha_{14}$ | $\\{13,13\\}$
$\\{13,13,16\\}$ | $\alpha_{14}$ | $\\{13,16\\}$
$\\{13,14,16\\}$ | $\alpha_{8}$ | $\\{13,14\\}$
$\\{13,14,17\\}$ | $\alpha_{8}$ | $\\{13,17\\}$
$\\{13,15,17\\}$ | $\alpha_{6}$ | $\\{13,15\\}$
$\\{13,15,18\\}$ | $\alpha_{6}$ | $\\{13,18\\}$
$\\{13,16,18\\}$ | $\alpha_{1}$ | $\\{13,16\\}$
$\\{13,16,19\\}$ | $\alpha_{1}$ | $\\{13,19\\}$
$\\{13,17,19\\}$ | $\alpha_{21}$ | $\\{13,19\\}$
$\\{13,18,19\\}$ | $\alpha_{21}$ | $\\{13,19\\}$
$\\{13,19,19\\}$ | $\alpha_{7}$ | $\\{13,19\\}$
$\\{13,19,20\\}$ | $\alpha_{5}$ | $\\{13,19\\}$
$\\{13,19,21\\}$ | $\alpha_{7}$ | $\\{13,19\\}$
$\\{13,19,22\\}$ | $\alpha_{5}$ | $\\{13,19\\}$
$\exp((G_{34},A_{3}),\kappa)=\\{13,19,23\\}$ |  |

Table 6. Induction tables for
${\mathscr{A}}=(G_{34},A_{1}^{2}),({\mathscr{A}}^{H},\kappa)=((G_{34},A_{1}A_{2}),\kappa),((G_{34},A_{3}),\kappa)$

The defining polynomial of $(G_{34},A_{2})$ is given by

$\displaystyle Q(G_{34},A_{2})=\ $
$\displaystyle(x_{3}-x_{4})(x_{3}-\zeta^{2}x_{4})(\zeta^{2}x_{1}+\zeta
x_{2}+x_{3}+3x_{4})(x_{1}-\zeta x_{2})(x_{3}-\zeta
x_{4})((-2\zeta-\zeta^{2})x_{4})(\zeta^{2}x_{1}+\zeta x_{2}+\zeta
x_{3}+(-2\zeta-\zeta^{2})x_{4})$
$\displaystyle(x_{1}+x_{2}+x_{3}+3x_{4})(\zeta^{2}x_{1}+\zeta
x_{2}+\zeta^{2}x_{3}+(-\zeta-2\zeta^{2})x_{4})(\zeta^{2}x_{1}+\zeta
x_{2}+x_{3})(x_{1}+x_{2}+\zeta x_{3}+(-2\zeta-\zeta^{2})x_{4})$
$\displaystyle(x_{1}+x_{2}+\zeta^{2}x_{3}+(-\zeta-2\zeta^{2})x_{4})(x_{1}+x_{2}+x_{3})(\zeta^{2}x_{1}+\zeta
x_{2}+\zeta^{2}x_{3}+(-\zeta+\zeta^{2})x_{4})(\zeta^{2}x_{1}+\zeta x_{2}+\zeta
x_{3}+(\zeta-\zeta^{2})x_{4})$
$\displaystyle(x_{1}+x_{2}+\zeta^{2}x_{3}+(-\zeta+\zeta^{2})x_{4})(x_{1}+x_{2}+\zeta
x_{3}+(\zeta-\zeta^{2})x_{4})(\zeta x_{1}+x_{2}+x_{3}+(\zeta-\zeta^{2})x_{4})$
$\displaystyle(\zeta^{2}x_{1}+\zeta^{2}x_{2}+x_{3}+(\zeta-\zeta^{2})x_{4})(-\zeta
x_{1}+x_{4})(-\zeta^{2}x_{2}+x_{4})(\zeta^{2}x_{1}+x_{2}+x_{3}+(-\zeta+\zeta^{2})x_{4})$
$\displaystyle(\zeta x_{1}+x_{2}+\zeta
x_{3}+(-\zeta-2\zeta^{2})x_{4})(\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta
x_{3}+(-\zeta-2\zeta^{2})x_{4})(\zeta^{2}x_{1}+x_{2}+\zeta^{2}x_{3}+(-2\zeta-\zeta^{2})x_{4})$
$\displaystyle(x_{1}+\zeta^{2}x_{2}+\zeta x_{3}+3x_{4})(\zeta
x_{1}+x_{2}+\zeta^{2}x_{3}+3x_{4})(\zeta x_{1}+\zeta x_{2}+\zeta
x_{3}+3x_{4})(\zeta^{2}x_{1}+\zeta^{2}x_{2}+\zeta^{2}x_{3}+3x_{4})(x_{1}-\zeta^{2}x_{3})$
$\displaystyle(x_{2}-\zeta
x_{3})(x_{1}-x_{3})(x_{2}-\zeta^{2}x_{3})(x_{1}+\zeta
x_{2}+\zeta^{2}x_{3}+3x_{4})(\zeta^{2}x_{1}+x_{2}+\zeta x_{3})(x_{1}-\zeta
x_{4})(x_{2}-x_{4})$ $\displaystyle(\zeta^{2}x_{1}+x_{2}+x_{3}+(-\zeta
2\zeta^{2})x_{4})(x_{1}-x_{4})(x_{2}+(-\zeta^{2})x_{4})(x_{1}+\zeta
x_{2}+x_{3}+(-2\zeta-\zeta^{2})x_{4})(x_{1}-\zeta x_{3})(x_{2}-x_{3})$
$\displaystyle(\zeta^{2}x_{1}+x_{2}+\zeta
x_{3}+3x_{4})(\zeta^{2}x_{1}+x_{2}+\zeta^{2}x_{3}+(\zeta-\zeta^{2})x_{4})(x_{1}+\zeta
x_{2}+\zeta x_{3}+(-\zeta+\zeta^{2})x_{4})(\zeta
x_{1}+\zeta^{2}x_{2}+x_{3}+3x_{4})$
$\displaystyle(x_{1}-x_{2})(x_{1}-\zeta^{2}x_{2}).$

Now the Ziegler restriction of $(G_{34},A_{2})$ with respect to
$H=\ker((-2\zeta-\zeta^{2})x_{4})$ gives $((G_{34},G(3,3,3)),\kappa)$ with
defining polynomial

$\displaystyle Q((G_{34},G(3,3,3)),\kappa)=\ $
$\displaystyle(x_{3})^{3}(\zeta^{2}x_{1}+\zeta x_{2}+x_{3})^{4}(x_{1}-\zeta
x_{2})(\zeta^{2}x_{1}+\zeta x_{2}+\zeta
x_{3})^{3}(x_{1}+x_{2}+x_{3})^{4}(\zeta^{2}x_{1}+\zeta
x_{2}+\zeta^{2}x_{3})^{3}(x_{1}+x_{2}+\zeta x_{3})^{3}$
$\displaystyle(x_{1}+x_{2}+\zeta^{2}x_{3})^{3}(-\zeta
x_{1})^{3}(-\zeta^{2}x_{2})^{3}(\zeta^{2}x_{1}+x_{2}+x_{3})^{3}(\zeta^{2}x_{1}+x_{2}+\zeta^{2}x_{3})^{3}(x_{1}-\zeta^{2}x_{3})(x_{2}-\zeta
x_{3})(x_{1}-x_{3})$ $\displaystyle(x_{2}-\zeta^{2}x_{3})(x_{1}+\zeta
x_{2}+\zeta^{2}x_{3})^{4}(x_{1}-\zeta
x_{3})(x_{2}-x_{3})(x_{1}-x_{2})(x_{1}-\zeta^{2}x_{2}).$

The Ziegler restriction of $(G_{34},A_{2})$ with respect to
$H=\ker(x_{3}-x_{4})$ gives $((G_{34},A_{3}),\kappa)$ with defining polynomial

$\displaystyle Q((G_{34},A_{3}),\kappa)=\ $
$\displaystyle((-\zeta-2\zeta^{2})x_{3})^{3}(\zeta^{2}x_{1}+\zeta
x_{2}+4x_{3})(x_{1}-\zeta x_{2})(\zeta^{2}x_{1}+\zeta
x_{2}+x_{3})^{3}(x_{1}+x_{2}+4x_{3})(x_{1}+x_{2}+x_{3})^{3}$
$\displaystyle(\zeta^{2}x_{1}+\zeta
x_{2}+(-\zeta+2\zeta^{2})x_{3})^{2}(\zeta^{2}x_{1}+\zeta
x_{2}+(2\zeta-\zeta^{2})x_{3})^{2}(x_{1}+x_{2}+(-\zeta+2\zeta^{2})x_{3})^{2}(x_{1}+x_{2}+(2\zeta-\zeta^{2})x_{3})^{2}$
$\displaystyle(\zeta
x_{1}+x_{2}+(-2\zeta^{2})x_{3})^{2}(\zeta^{2}x_{1}+\zeta^{2}x_{2}+(-2\zeta^{2})x_{3})^{2}(-\zeta
x_{1}+x_{3})^{2}(-\zeta^{2}x_{2}+x_{3})^{2}(\zeta^{2}x_{1}+x_{2}-2\zeta
x_{3})^{2}(x_{1}-x_{3})^{2}$
$\displaystyle(x_{2}-\zeta^{2}x_{3})^{2}(x_{1}+\zeta
x_{2}+(-3\zeta-2\zeta^{2})x_{3})^{2}(\zeta^{2}x_{1}+x_{2}+\zeta
x_{3})^{3}(x_{1}-\zeta x_{3})^{2}(x_{2}-x_{3})^{2}$
$\displaystyle(\zeta^{2}x_{1}+x_{2}+(-2\zeta-3\zeta^{2})x_{3})^{2}(\zeta
x_{1}+\zeta^{2}x_{2}+4x_{3})(x_{1}-x_{2})(x_{1}-\zeta^{2}x_{2}).$

$\exp({\mathscr{A}}^{\prime},\mu^{\prime})$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp((G_{34},A_{3}),\mathbbm{1})=\\{1,11,13\\}$ | $\alpha_{6}$ | $\\{11,13\\}$
$\\{2,11,13\\}$ | $\alpha_{6}$ | $\\{11,13\\}$
$\\{3,11,13\\}$ | $\alpha_{15}$ | $\\{11,13\\}$
$\\{4,11,13\\}$ | $\alpha_{11}$ | $\\{11,13\\}$
$\\{5,11,13\\}$ | $\alpha_{12}$ | $\\{11,13\\}$
$\\{6,11,13\\}$ | $\alpha_{4}$ | $\\{11,13\\}$
$\\{7,11,13\\}$ | $\alpha_{4}$ | $\\{11,13\\}$
$\\{8,11,13\\}$ | $\alpha_{1}$ | $\\{11,13\\}$
$\\{9,11,13\\}$ | $\alpha_{1}$ | $\\{11,13\\}$
$\\{10,11,13\\}$ | $\alpha_{19}$ | $\\{11,13\\}$
$\\{11,11,13\\}$ | $\alpha_{19}$ | $\\{11,13\\}$
$\\{11,12,13\\}$ | $\alpha_{22}$ | $\\{12,13\\}$
$\\{12,12,13\\}$ | $\alpha_{21}$ | $\\{12,13\\}$
$\\{12,13,13\\}$ | $\alpha_{20}$ | $\\{12,13\\}$
$\\{12,13,14\\}$ | $\alpha_{18}$ | $\\{13,14\\}$
$\\{13,13,14\\}$ | $\alpha_{17}$ | $\\{13,14\\}$
$\\{13,14,14\\}$ | $\alpha_{16}$ | $\\{13,14\\}$
$\\{13,14,15\\}$ | $\alpha_{14}$ | $\\{13,15\\}$
$\\{13,15,15\\}$ | $\alpha_{13}$ | $\\{13,15\\}$
$\\{13,15,16\\}$ | $\alpha_{10}$ | $\\{13,16\\}$
$\\{13,16,16\\}$ | $\alpha_{9}$ | $\\{13,16\\}$
$\\{13,16,17\\}$ | $\alpha_{8}$ | $\\{13,16\\}$
$\\{13,16,18\\}$ | $\alpha_{7}$ | $\\{13,16\\}$
$\exp((G_{34},A_{3}),\kappa)=\\{13,16,19\\}$ |  |

$\exp({\mathscr{A}}^{\prime},\mu^{\prime})$ | $\alpha_{H}$ | $\exp({\mathscr{A}}^{\prime\prime},\mu^{*})$
---|---|---
$\exp((G_{34},G(3,3,3)),\mathbbm{1})=\\{1,7,13\\}$ | $\alpha_{12}$ | $\\{7,13\\}$
$\\{2,7,13\\}$ | $\alpha_{11}$ | $\\{7,13\\}$
$\\{3,7,13\\}$ | $\alpha_{10}$ | $\\{7,13\\}$
$\\{4,7,13\\}$ | $\alpha_{9}$ | $\\{7,13\\}$
$\\{5,7,13\\}$ | $\alpha_{8}$ | $\\{7,13\\}$
$\\{6,7,13\\}$ | $\alpha_{7}$ | $\\{7,13\\}$
$\\{7,7,13\\}$ | $\alpha_{6}$ | $\\{7,13\\}$
$\\{7,8,13\\}$ | $\alpha_{4}$ | $\\{7,13\\}$
$\\{7,9,13\\}$ | $\alpha_{1}$ | $\\{7,13\\}$
$\\{7,10,13\\}$ | $\alpha_{4}$ | $\\{10,13\\}$
$\\{8,10,13\\}$ | $\alpha_{7}$ | $\\{10,13\\}$
$\\{9,10,13\\}$ | $\alpha_{8}$ | $\\{10,13\\}$
$\\{10,10,13\\}$ | $\alpha_{9}$ | $\\{10,13\\}$
$\\{10,11,13\\}$ | $\alpha_{10}$ | $\\{10,13\\}$
$\\{10,12,13\\}$ | $\alpha_{12}$ | $\\{10,13\\}$
$\\{10,13,13\\}$ | $\alpha_{11}$ | $\\{10,13\\}$
$\\{10,13,14\\}$ | $\alpha_{6}$ | $\\{10,13\\}$
$\\{10,13,15\\}$ | $\alpha_{1}$ | $\\{10,13\\}$
$\\{10,13,16\\}$ | $\alpha_{17}$ | $\\{13,16\\}$
$\\{11,13,16\\}$ | $\alpha_{17}$ | $\\{13,16\\}$
$\\{12,13,16\\}$ | $\alpha_{17}$ | $\\{13,16\\}$
$\\{13,13,16\\}$ | $\alpha_{5}$ | $\\{13,16\\}$
$\\{13,14,16\\}$ | $\alpha_{5}$ | $\\{13,16\\}$
$\\{13,15,16\\}$ | $\alpha_{2}$ | $\\{13,16\\}$
$\\{13,16,16\\}$ | $\alpha_{2}$ | $\\{13,16\\}$
$\\{13,16,17\\}$ | $\alpha_{5}$ | $\\{13,16\\}$
$\\{13,16,18\\}$ | $\alpha_{2}$ | $\\{13,16\\}$
$\exp((G_{34},G(3,3,3)),\kappa)=\\{13,16,19\\}$ |  |

Table 7. Induction tables for
${\mathscr{A}}=(G_{34},A_{2}),({\mathscr{A}}^{H},\kappa)=((G_{34},A_{3}),\kappa),((G_{34},G(3,3,3)),\kappa)$

Secondly, we checked that in each instance in Table 4 when ${\mathscr{A}}$ is
of rank $5$, none of the $({\mathscr{A}}^{\prime\prime},\kappa)$ is
inductively free. Specifically, we proved a slightly stronger statement,
namely that none of the multiarrangements $((G_{34},A_{1}^{2}),\kappa)$ and
$((G_{34},A_{2}),\kappa)$ is additively free, cf. Definition 2.14. This was
accomplished by computer calculations carried out in SAGE [St15] as follows:
If $({\mathscr{A}},\kappa)$ were additively free with a chain of free
multiplicities $({\mathscr{A}},\mu_{i})$ starting at the empty
$4$-arrangement, ending at $({\mathscr{A}},\kappa)$, then according to Theorem
2.9, for each Euler restriction with respect to $H_{i}$ say, in the chain
$({\mathscr{A}},\mu_{i})$, we get
$\exp({\mathscr{A}}^{H_{i}},\mu_{i}^{*})\subset\exp({\mathscr{A}},\mu_{i})$ at
every step. In order to show that $({\mathscr{A}},\kappa)$ does not admit such
a chain, we start with the free arrangement $({\mathscr{A}},\kappa)$ along
with its set of exponents and successively eliminate hyperplanes. Starting
with the exponents of $({\mathscr{A}},\kappa)$, we assign “virtual exponents”
to a deletion along such a chain recursively as follows. We determine the
number of hyperplanes in the corresponding Euler restriction
$({\mathscr{A}}^{H_{i}},\mu_{i}^{*})$ for any $H_{i}\in{\mathscr{A}}$. Theorem
2.9 can only be employed if $|\mu_{i}^{*}|$ equals the sum of $3$ out of $4$
“virtual exponents” of $({\mathscr{A}},\mu_{i})$. So if we do have such an
equality, we save the corresponding deletion
$({\mathscr{A}}^{\prime},\mu_{i}^{\prime})=({\mathscr{A}},\mu_{i-1})$ in a
list, lower the “virtual exponent” that was not part of the sum by $1$ and
assign this new set of “virtual exponents” in the next round of our algorithm
to $({\mathscr{A}},\mu_{i-1})$. We do stress the “virtual exponents” that are
being determined along the way need not be actual exponents of
$({\mathscr{A}},\mu_{i-1})$, as the latter may not be free. Once every
hyperplane $H_{i}\in{\mathscr{A}}_{i}$ has been tested, we repeat the
algorithm with each of the deletions $({\mathscr{A}},\mu_{i-1})$ along with
its set of “virtual exponents” that was incurred in the previous round.

It turns out that, as a result of this algorithm, every chain of
subarrangements $({\mathscr{A}},\mu_{i})$ admitting a correct number of
hyperplanes in each restriction such that Theorem 2.9 could in principal be
applied does terminate well before reaching the empty arrangement.
Consequently, there is no free chain of subarrangements for
$({\mathscr{A}},\kappa)$, so in particular, $({\mathscr{A}},\kappa)$ fails to
be additively free, so it is not inductively free. ∎

Acknowledgments: This work was supported by DFG-Grant RO 1072/21-1 (DFG
Project number 494889912) to G. Röhrle.

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