数理解析研究所講究録
第 1689 巻 2010 年 89-100

89

Primitive derivation, Coxeter
multiarrangements and some examples
Takuro Abe*

Abstract
This is a survey on recent developments about Coxeter multiarrangements, its freeness and the relation with the primitive derivation.

Also, we give some examples on free and non-free Coxeter multiarrangements.

Introduction$\prime o$a
Let $V=V^{\ell}$ be an -dimensional vector space over a field.
and $S:=$ Sym
basis for the dual vector space. Let Der $(S)$ denote the S-module:Fix an inner productthe S-module of differential l-forms, i.e.,of -linear derivations of and. Also, for the quotient fieldDerand$F$ of$F$, Der
denote the derivation and differential modules with
and
$F$
:
coefficients in . Note that can be canonically extended to
$F$ .
$(F)$
(resp.
Der
A non-zero element)
zero or homogeneous of(resp.
isis homogeneous of degree if
degree for each .
A hyperplane armngement (or simply an arrangement) is a finite colleccontains the origin,tion of affine hyperplanes in $V$ . If each hyperplane in
we say that is central. In this article we assume that all arrangements arecentral unless otherwise specified. A multiplicity $m$ on an arrangement isis called a multiarrangement. Let
and a paira map$|m|$ denote the sum of the multiplicities. When $m\equiv 1,$
and sometimes called a simple
is the same as the hyperplane arrangementsuch
fix a linear form
arrangement. For each hyperplane$\ell$$\mathbb{R},$$(V^{*})\otimes \mathbb{C}\simeq \mathbb{C}[x_{1}, \ldots, x_{l}]$$V^{*}$$I^{*}$$V^{*}\cross V^{*}arrow \mathbb{R}$$S$$\mathbb{C}$$(S)=\oplus_{i=1}^{\ell}S\cdot\partial_{x_{i}}$$S$$\{x_{1}, \ldots, x_{\ell}\}$$\Omega_{V}^{1}$$:=\oplus_{i=1}^{\ell}S\cdot dx_{i}$$\Omega_{S}^{1}$$\Omega_{F}$$I^{*}$$I^{*}$$\omega=\sum_{i=1}^{\ell}g_{i}dx_{i}\in$$\theta=\sum_{i=1}^{\ell}f_{i}\partial_{x_{i}}\in$$\Omega_{F}^{1})$$p$$\Omega_{F}^{1}\cross\Omega_{F}^{1}arrow$$f_{i}$$g_{i}$$i$$p$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$m:\mathcal{A}arrow \mathbb{Z}_{\geq 0}$$(\mathcal{A}, m)$$\sum_{H\in \mathcal{A}}m(H)$$(\mathcal{A}, m)$$\mathcal{A}$$H\in \mathcal{A}$$\alpha_{H}\in V^{*}$*Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-Cho, Kyoto
606-8502, Japan. email:abetaku@math.kyoto-u.ac.jp.

90that $ker(\alpha_{H})=H$ . Putdenote the constant multiplicity on$Q(\mathcal{A}, m)$$:= \prod_{H\in \mathcal{A}}\alpha_{H}^{m(H)}$$\mathcal{A}$. Also, for$k\in \mathbb{Z}$

let

$k$

:

$k(H)=k(\forall H\in \mathcal{A})$.

The main objects in this article are the logari thmic dentvation moduleofdefined by$D(\mathcal{A}, m)$$(\mathcal{A}, m)$$D(\mathcal{A}, m):=\{\theta\in$

Der$(S)|\theta(\alpha_{H})\in S\cdot\alpha_{H}^{m(H)}$and the logarithmic differential module$\Omega^{1}(\mathcal{A}, m)$

$:=$

{

$\Omega^{1}(\mathcal{A}, m)$$\omega\in\frac{1}{Q(\mathcal{A},m)}\Omega_{V}^{1}|d\alpha_{H}\wedge\omega$(for all$H\in \mathcal{A})\}$$(\mathcal{A}, m)$defined byofis regular along $H$ for all,$H\in \mathcal{A}$}.

It is well-known thatandare S-dual modules, and hence
reflexive in general. A multiarrangement
is free if
is a free
S-module of rank . If
is free, then there exists a homogeneous freebasis
. Then we define the exponents of a free multifor$:=(\deg(\theta_{1}), \ldots, \deg(\theta_{\ell}))$ . The exponents
arrangementbyare independent of a choice of a basis. When $m\equiv 1,$
is denoted.
by$D(\mathcal{A}, m)$$\Omega^{1}(\mathcal{A}, m)$$(\mathcal{A}, m)$$D(\mathcal{A}, m)$$(\mathcal{A}, m)$$\ell$$D(\mathcal{A}, m)$$\{\theta_{1}, \ldots, \theta_{\ell}\}$$(\mathcal{A}, m)$$\exp(\mathcal{A}, m)$$\exp(\mathcal{A}, 1)$$\exp(\mathcal{A})$A multiarrangement was introduced by Ziegler in [23], and after that,
there have been a lot of studies and results related to it. Especially, Yoshinaga’s two works [20] and [21] have made the study of multiarrangements
and their freeness more interesting and important. For example, [6] generalized a characteristic polynomial and [7] the addition-deletion theorems for
multiarrangements. Also, the totally non-freeness of generic arrangements
was proved in [22], the first complete classification of an arrangement which
admits both free and non-free multiplicities was done in [1], and the geometric characterization of totally free arrangements was obtained in [8]. Morerecently, the author generalized the definition of multiarrangements to thatwith multiplicities consisting of positive and negative integers, and gave adefinition of a logarithmic module associated to them in [2]. In other words,a (genemlized) multiplicity is a function
and the corresponding
logarithmic module
is defined by$m:\mathcal{A}arrow \mathbb{Z}$$D\Omega(\mathcal{A}, m)$$D \Omega(\mathcal{A}, m):=\frac{D(\mathcal{A}_{+},m_{+})}{Q_{-}}\cap I^{*}(\Omega^{1}(\mathcal{A}_{-}, -m_{-}))$.

For details, see [2], in which a theory of generalized multiarrangements andtheir logarithmic modules had been constructed. For example, the freeness,

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exponents, reflexivity (duality) and Saito) criterion hold true foras for usual multiarrangements, and a duality of Coxeter multiarrangements
was generalized by using this module and Kyoji Saito’s primitive derivation.
As we could see in the above, the theory of free multiarrangements andits investigations are developing very rapidly, and producing several new
concepts and ideas which can be used in the study of simple arrangements.
Among them, one of the most important roles is played by the research of
Coxeter multiarrangements, initiated by Solomon and Terao in [15], developed by Terao in [16], and still being studied intensively by Terao, Wakamiko,
Wakefield, Yuzvinsky, Yoshinaga and the author. This note is devoted forthe survey of these studies and results with some new examples which are
related to free Coxeter arrangements.
$D\Omega(\mathcal{A}, m)$$s$

Acknowledements. The author thanks Professor Hideaki Morita for theinvitation to the RIMS conference “Representation theory and Combinatorics” The author is supported by Grant in Aid for Young Scientists (No.
21740014).

1

Coxeter multiarrangements

In this section we review results on free Coxeter multiarrangements. It has
been very difficult to determine whether given multiarrangements are free ornot, even for simple arrangements. One of the reasons why this problem isdifficult is there are only few tools or methods to study the freeness. Though
we have addition-deletion theorems ([7]) and characteristic polynomials for
non-freeness criterion ([6]), there are no sufficient freeness criterions so far.
On the other hand, if we go back to the original point of free arrangements,
we can obtain very strong theory for the freeness, i.e., the invariant theory of
Coxeter groups. To see it, recall the most important theorem by Chevalley:
Theorem 1.1 ([10])For the invarian ring $R$ $:=S^{W}$ , there exis $thom$ ogeneous basic in varian $ts$
such th at$t$$P_{1},$$\ldots,$$P_{\ell}\in R$$R=S^{W}=\mathbb{R}[P_{1}, \ldots, P_{\ell}]$.

Moreover, if we put $\deg P_{i}=m_{i}+1$ with $m_{1}\leq m_{2}\leq\cdots\leq m_{\ell}$ , then$\exp(W)=(m_{1}, \ldots, m_{\ell})$and $1=m_{1}<m_{2}\leq\cdots\leq m_{\ell-1}<m_{\ell}=h-1$ with th Coxeter number
$W$ .
$e$

$h$

of

92
The following is due to Kyoji Saito, which is the starting point of freearrangement theory.

Theorem 1.2 (K. Saito)
form an W-invariant basis
For a Coxeter arrangement$d\{IdP_{1},\ldots,
IdP_{\ell}\}$. In articular,
form that for, anfor$\mathcal{A},$$\{I^{*}\partial_{P_{1}}, \ldots, I^{*}\partial_{P_{\ell}}\}$$\Omega^{1}(\mathcal{A}, 1)$$s$$D(\mathcal{A}, 1)$$s$$\exp(\mathcal{A})=\exp(W)$$p$.

Also, K. Saito gives us a very interesting derivation for the research of
free Coxeter arrangements.

Definition 1.3 (K. Saito)
The lowest degree derivation
primitive derivation.

$D:=\partial_{P_{l}}\in$

Der $(R)$ (up to scalors) is called a

The primitive derivation will play, in the rest of this article, a key role
to construct free bases for Coxeter multiarrangements. Also, the primitive
derivation is essential to construct the Hodge filtration (see [14] for example),and primitive filtration ([4] and [5]), which we do not see in this article.
Now it is natural to consider the freeness of Coxeter multiarrangements.
The first research on this problem is done by Solomon and Terao in [15].

Theorem 1.4 ([15])Letbe a Coxeter arrangement with the Coxeter number
is free with $\exp(\mathcal{A}, 2)=(h, \ldots, h)$ .
multiarrangement$\mathcal{A}$$h$. Then the$(\mathcal{A}, 2)$Theorem 1.4 may seem to be a very special case. However, the following
work by Terao shows that Theorems 1.2 and 1.4 are very essential in thetheory of free Coxeter multiarrangements.

Theorem 1.5 ([16])Letbe a Coxeter arrangement with the Coxeter number .
$h$$\mathcal{A}$(1) The multiarrangement$(\mathcal{A}, 2k)$

is free with $\exp(\mathcal{A}, 2k)=(kh, \ldots, kh)$ .

(2) The multiarrangement $(\mathcal{A}, 2k+1)$ is free with $\exp(\mathcal{A}, 2k+1)=(kh+$$m_{1},$$\ldots,$$kh+m_{\ell})$.

Combining these theorems, we can see that there seems to be a shiftingof exponents and freeness. The shifting phenomenon will be studied soonwith more strong statement. We note that the results above are provedby constructing explicit bases by using the invariant theory and primitive
derivations.

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2

Shifting isomorphism

In this section we introduce a different method from that in the last section
for constructing bases for Coxeter multiarrangements.
This section aims at constructing an isomorphism$\Phi_{k}:D(\mathcal{A}, m)arrow D(\mathcal{A}, 2k+m)$,

which gives the shifting of multiplicity $2k$ . Before explaining how to construct, we concentrate our interest on the starting module. We use
the multiplicity $m$ : $\mathcal{A}arrow\{+1,0, -1\}$ with the logarithmic module
defined in the introduction.
$D(\mathcal{A}, m)$$D\Omega(\mathcal{A}, m)$

Remark 2.1
However, even for a generalized $m$ ultiplicity $m$ :, we use the notationnotin this article, different from the origin $al$ paper [2].
$\mathcal{A}arrow \mathbb{Z}$$D(\mathcal{A}, m)$$D\Omega(\mathcal{A}, m)$

First for the construction, we have to find a (multi-Euler” derivation. To
construct a basis for the logarithmic derivation module, the Euler derivation
plays the key role in the sense thatis tangent to any
hyperplanes with the multiplicity one:$\theta_{E};=\sum_{i=1}^{\ell}x_{i}\partial_{x_{i}}$$\theta_{E}$.

$\theta_{E}(\alpha_{H})=\alpha_{H}\in S\cdot\alpha_{H}\backslash S\cdot\alpha_{H}^{2}(\forall\alpha_{H}\in V^{*})$

This is the special derivation, and we cannot expect the existence of sucha kind of derivations for multiarrangements. However, Yoshinaga noticed
that we can find a derivation similar to the Euler derivation associated to
the fixed Coxeter multiarrangements. Note that the primitive derivation $D$
induces the $T$ $:=ker(D : Rarrow R)-$ isomorphism: Der $(S^{W})arrow$ Der $R$( $e.g.$ , see [14]) and$\nabla_{D}$...

$\subset D(\mathcal{A}, 2k+1)^{W}\subset D(\mathcal{A}, 2k-1)^{W}\subset\cdots\subset D(\mathcal{A}, 1)^{W}\subset$

Der $R$ .

Thus we can always define the derivation. Thenthe following fact, proved by Yoshinaga in [19] and generalized in [9], is very
important.
$E_{k}$$:=\nabla_{D}^{-k}\theta_{E}\in D(\mathcal{A})^{W}$

Proposition 2.2
The derivation $E_{k}\in D(\mathcal{A}, 2k+1)^{W}$ .

Hence we can use
us construct a map$E_{k}$

as a kind of the “multi-Euler” derivation. Now let$\Phi=\Phi_{k}$by$D(\mathcal{A}, m)\ni\theta\mapsto\Phi_{k}(\theta):=\nabla_{\theta}E_{k}$.

94
To prove that$\Phi_{k}$gives rise to an S-module isomorphism$D(\mathcal{A}, m)\simeq D(\mathcal{A}, 2k+m)$,

we have to check the following three points:
is well-defined.
Point 1.
Point 2.
is injective.
Point 3.
is surjective.
We do not explain details on the above three. For details see [2]. We justpoint out what plays the key role to check above three points.
$\Phi_{k}$$\Phi_{k}$$\Phi_{k}$and count
Point 1. To prove this point, we use the W-invariance ofcarefully.
orders of poles along
Point 2. This part can be proved by using the totally same argument as in
9 , which is essentially proved in 16 .
is free. Then,
Point 3. We prove in two steps. First we prove whenis free, we can use the structure of this free module to prove
since
.
, for
a general$E_{k}$$H\in \mathcal{A}$

$[$

$[$

$]$

$]$

$(\mathcal{A}, m)$$D(\mathcal{A}, -1)$$(\mathcal{A}, m)$$D(\mathcal{A}, m)\subset D(\mathcal{A}, -1)$

Summarizing these, we obtain the following:

Theorem 2.3 ([2])
The $map\Phi_{k}$ gives rise to an S-module isomorphism$\Phi_{k}:D(\mathcal{A}, m)arrow D(\mathcal{A}, 2k+m)$

In particular, for a freewith basis$D(\mathcal{A}, m)$with basis$\nabla_{\theta_{1}}E_{k},$$\ldots,$$\theta_{1},$$\nabla_{\theta_{\ell}}E_{k}$.

$\theta_{\ell},$$\ldots,$$D(\mathcal{A}, 2k+m)$

is free.

Remark 2.4
itself forThough Theorem 2.3 gives an explicit form, to compute$m:\mathcal{A}arrow\{+1,0, -1\}$ is not easy even for a Coxeter arrangemen $ts$ of type
. We will see it in the next section.
$D(\mathcal{A}, m)$$A_{3}$

3

Remarks and Examples

In the previous section we restricted our interest on multiplicities$\{+1,0, -1\}$ . Then it is natural to ask whether these multiplicities can beextended or not. For example, do the results in the previous section holdtrue for multiplicities like $m:\mathcal{A}arrow\{+2, +1,0, -1, -2\}$ ? The answer is NO,
see the following example:$m:\mathcal{A}arrow$

95
Example 3.1$Con$ sider a Coxeter arrangementof $typeA_{3}$ defined by $xyz(x-y)(x-$$z)(y-z)=0$ . Note th at we use a notation $x_{1}=x,$ $x_{2}=y,$ $x_{3}=z,$ $x_{4}=w$ inthis section. Consider th following two $mul$ tiarrangemen $ts$ :$\mathcal{A}$$e$$(\mathcal{A}, m_{1}):x^{2}y^{2}z^{2}(x-y)^{2}(x-z)^{0}(y-z)^{0}=0$,$(\mathcal{A}, m_{2}):x^{4}y^{4}z^{4}(x-y)^{4}(x-z)^{2}(y-z)^{2}=0$.

These multipliciti es on can be expressed in the same way as in the previous
section with $m$ : $\mathcal{A}arrow\{+2, +1,0, -1, -2\}$ . If the same result holds true, thenandhave to be isomorphic, but they do not. In fact, we
can prove th following;$\mathcal{A}$$D(\mathcal{A}, m_{1})$$D(\mathcal{A}, m_{2})$$e$Lemma 3.2
A multiarrangement$(\mathcal{A}, m)$defined by$x^{2k}y^{2k}z^{2k}(x-y)^{2k}(x-z)^{2k-2}(y-z)^{2k-2}=0(k\in \mathbb{Z}_{\geq 1})$

is free if and only if $k=1$ .
Hen ce the condition on multiplicities is essential.
Proof. When $k=1$ then the multiarrangement is supersolvable in the senseof [7], hence free. Assume that $k>1$ andis free. Compute theupper bound of GMP, global mixed product. Note that $|m|=12k-4$ .
Hence the upper bound of $GMP(\mathcal{A}, m)$ is attained when the exponents are$(4k-2,4k-1,4k-1)$ . Hence$(\mathcal{A}, m)$$GMP(\mathcal{A}, m)\geq 2(4k-2)(4k-1)+(4k-1)^{2}=48k^{2}-32k+5$ .

Next let us compute the LMP, local mixed product. It is easy to get, when$k>1$ ,
$LMP(\mathcal{A}, m)=48k^{2}-32k+6>GML(\mathcal{A}, m)$ ,which is a contradiction.

$\square$

It seems to the author that the condition on multiplicities occurs fromwith thethe invariant theory. More explicitly, for a Coxeter arrangement$W$, it holds that
Coxeter group$\mathcal{A}$$D(\mathcal{A}, 2k)\subsetneq D(\mathcal{A}, 2k+1)$,

but$D(\mathcal{A}, 2k)^{W}=D(\mathcal{A}, 2k+1)^{W}$.

96
Hence the result in the previous section is to investigate the derivation modules which vanish in the invariant part. Also,$D(\mathcal{A}, -2k)\supsetneq D(\mathcal{A}, -2k+1)$,

but$D(\mathcal{A}, -2k)^{W}=D(\mathcal{A}, -2k+1)^{W}$.

with $m$ : $\mathcal{A}arrow\{+1,0, -1\}$ is a mixed module of these two
Since
modules, such a multiplicity seems not to be extended.
$D(\mathcal{A}, m)$

Let us give more examples of free bases for the -type arrangement witha non-constant $m:\mathcal{A}arrow\{+1, -1\}$ . Bases for constant multiplicities can beobtained by the invariant theory.
First consider the multiarrangement defined by$A_{3}$$\frac{(y-z)(z-x)(x-w)(y-w)(z-w)}{(x-y)}=0$ .

By [3] and [2] this multiarrangement is free with exponents $(0,1,2,1)$ . We
can obtain the explicit basis as follows:,$\theta_{0}$$=$$\partial_{x}+\partial_{y}+\partial_{z}.+\partial_{w}$$\theta_{1}$$=$$x\partial_{x}+y\partial_{y}+z\partial_{z}+w\partial_{w}$$\theta_{2}$$=$$x^{2}\partial_{x}+y^{2}\partial_{y}+z^{2}\partial_{z}+w^{2}\partial_{w}$$\theta_{3}$

$=$

,
,

$(x-z)(x-w)\partial_{x}-(y-z)(y-w)\partial_{y}$$x-y$Second consider the arrangement defined by$\frac{(x-y)(y-z)(z-x)(x-w)}{(y-w)(z-w)}=0$ .

By [3] and [2] this multiarrangement is free with exponents $(0,1,0,1)$ . We
can obtain the explicit basis as follows:,$\theta_{0}$$=$$\partial_{x}+\partial_{y}+\partial_{z}+\partial_{w}$$\theta_{1}$$=$$x\partial_{x}+y\partial_{y}+z\partial_{z}+w\partial_{w}$$\theta_{2}$$=$$(x-w)\partial_{w}$$\theta_{3}$$=$$\partial_{x}+\frac{x-w}{y-w}\partial_{y}+\frac{x-w}{z-w}\partial_{z}+\frac{-w^{2}-xy-xz+2xw+yz}{(y-w)(z-w)}\partial_{w}$,,.

The basis of a multiarrangement defined by$\frac{(y-z)(y-w)(z-w)}{(x-y)(x-z)(x-w)}=0$

97
can be found in [2].
Next consider the arrangement defined by$\frac{(x-y)(x-z)(x-w)}{(y-z)(y-w)(z-w)}=0$.

By [3] and [2] this multiarrangement is free with exponents $(0,1,0, -1)$ . We
can obtain the explicit basis as follows:,$\theta_{0}$$=$$\partial_{x}+\partial_{y}+\partial_{z}+\partial_{w}$$\theta_{1}$$=$$x\partial_{x}+y\partial_{y}+z\partial_{z}+w\partial_{w}$$\theta_{2}$$=$$\frac{x-y}{(y-z)(y-w)}\partial_{y}-\frac{x-z}{(y-z)(z-w)}\partial_{z}+\frac{x-w}{(y-w)(z-w)}\partial_{w}$$\theta_{3}$$=$$\frac{(x-y)(z-w)y\partial_{y}-(x-z)(y-w)z\partial_{z}+(x-w)(y-z)w\partial_{w}}{(y-z)(y-w)(z-w)}$,,.

Next consider the arrangement defined by$\frac{(x-z)(x-w)}{(x-y)(y-z)(y-w)(z-w)}=0$.

By [3] and [2] this multiarrangement is free with exponents $(0, -1,0, -1)$ .
We can obtain the explicit basis as follows:,$\theta_{0}$$=$$\partial_{x}+\partial_{y}+\partial_{z}+\partial_{w}$$\theta_{1}$$=$$\frac{1}{x-y}\partial_{x}-(\frac{1}{x-y}-\frac{1}{y-z}-\frac{1}{y-w})\partial_{y}-\frac{1}{y-z}\partial_{z}-\frac{1}{y-w}\partial_{w}$$\theta_{2}$$=$$\frac{x-y}{(y-z)(y-w)}\partial_{y}-\frac{x-z}{(y-z)(z-w)}\partial_{z}+\frac{x-w}{(y-w)(z-w)}\partial_{w}$$\theta_{3}$$=$$\frac{(x-y)(z-w)y\partial_{y}-(x-z)(y-w)z\partial_{z}+(x-w)(y-z)w\partial_{w}}{(y-z)(y-w)(z-w)}$

Finally consider the arrangement defined by$\frac{x-w}{(x-y)(x-z)(y-z)(y-w)(z-w)}=0$ .

,

,

.

98

By [3] and [2] this multiarrangement is free with exponents $(0, -1, -1, -2)$ .
We can obtain the explicit basis as follows:,$\theta_{0}$$=$$\partial_{x}+\partial_{y}+\partial_{z}+\partial_{w}$$\theta_{1}$$=$$\frac{1}{x-y}\partial_{x}-(\frac{1}{x-y}-\frac{1}{y-z}-\frac{1}{y-w})\partial_{y}-\frac{1}{y-z}\partial_{z}-\frac{1}{y-w}\partial_{w}$$\theta_{2}$$=$$\frac{x-y}{(y-z)(y-w)}\partial_{y}-\frac{x-z}{(y-z)(z-w)}\partial_{z}+\frac{x-w}{(y-w)(z-w)}\partial_{w}$$\theta_{3}$$=$$\frac{1}{Q_{-}}\{(y-z)(y-w)(z-w)\partial_{x}+(x-z)(z-w)(x-2y+w)\partial_{y}$,,$+(x-y)(y-w)(-x+2z-w)\partial_{z}+(x-y)(x-z)(y-z)\partial_{w}\}$ ,where$Q_{-:=}(x-y)(x-z)(y-z)(y-w)(z-w)$ .

Referencesar[1] T. Abe, Free and non-free multiplicity on the deleted
rangement. Proc. Japan Acad. Ser. A 83 (2007), No. 7, 99103.
$A_{3}$[2] T. Abe, A generalized logarithmic module and duality of Coxeter multiarrangements. arXiv.0807.2552vl.

[3] T. Abe, K. Nuida and Y. Numata, Signed-eliminable graphs
and free multiplicities on the braid arrangement. J. London
Math. Soc. 80 (2009), no. 1, 121-134.
[4] T. Abe and H. Terao, A primitive derivation and logarithmic differential forms of Coxeter arrangements. To appear in Math. Z., DOI: 10.1007/s00209-009-0489-8 (online).
arXiv: 0810. 3107v2, 2008
[5] T. Abe and H. Terao, A primitive filtration of the modules of invariant logarithmic forms of Coxeter arrangements.
arXiv.0910. 2506vl.

[6] T. Abe, H. Terao and M. Wakefield, The characteristic polynomial of a multiarrangement. Adv. in Math. 215 (2007), 825838.

99[7] T. Abe, H. Terao and M. Wakefield, The Euler multiplicity
and addition-deletion theorems for multiarrangements. J. London Math. Soc. 77 (2008), no. 2, 335-348.
[8] Totally free arrangements of hyperplanes. Proc. Amer. Math.
Soc. Vol. 137, no. 4, pp 1405-1410, 2009.

[9] T. Abe and M. Yoshinaga, Coxeter multiarrangementswith quasi-constant multiplicities. To appear in J. Algebra,
DOI: 10.1016/j.jalgebra.2009.07.025 (online). arXiv:0708.3228.

[10] C. Chevalley, Invariants of finite groups generated by reflections. Amer. J. Math. 77 (1955), 778-782.
[11] P. Orlik and H. Terao, Armngements of hyperplanes.
Grundlehren der Mathematischen Wissenschaften, 300.
Springer-Verlag, Berlin, 1992.

[12] K. Saito, On the uniformization of complements of discriminant loci. In: Conference Notes. Amer. Math. Soc. Summer
Institute, Williamstown, 1975.
[13] K. Saito, On a linear structure of the quotient variety by afinite reflection group. Publ. RIMS, Kyoto Univ. 29 (1993),

535-579.
[14] K. Saito, Uniformization of the orbifold of a finite reflectiongroup. RIMS preprint 1414 (2003).
[15] L. Solomon and H. Terao, The double Coxeter arrangements.
Comment. Math. Helv. 73 (1998), 237-258.

[16] H. Terao, Multiderivations of Coxeter arrangements. Invent.
Math. 148 (2002), 659-674.

[17] H. Terao, The Hodge filtration and the contact-order filtrationof derivations of Coxeter arrangements. Manuscripta Math.
118 (2005), 1-9.
[18] A. Wakamiko, On the Exponents of 2-Multiarrangements.
Tokyo J. Math. 30 (2007), no. 1, 99-116.

[19] M. Yoshinaga, The primitive derivation and freeness of multiCoxeter arrangements. Proc. Japan Acad. Ser. A 78 (2002),
no. 7, 116-119.

100[20] M. Yoshinaga, Characterization of a free arrangement andconjecture of Edelman and Reiner, Invent. Math., 157 (2004),
no.2, 449-454.
[21] M. Yoshinaga, On the freeness of 3-arrangements. Bull. London. Math. Soc. 37 (2005), no. 1, 126-134.

[22] M. Yoshinaga, On the extendability of free multiarrangements.
arXiv:0710.5044. To appear in the proceeding of 2007 CIMPA
Summer School (Arrangements, Local systems and Singularities”[23] G. M. Ziegler, Multiarrangements of hyperplanes and theirfreeness. in Singularities (Iowa City, IA, 1986), 345-359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989.