数理解析研究所講究録
第 1795 巻 2012 年 135-143

135

A BASIS FOR THE MODULE OF DIFFERENTIALOPERATORS OF ORDER 2 ON THE BRAID

HYPERPLANE ARRANGEMENTNORIHIRO NAKASHIMA

ABSTRACT. The braid arrangement is one of most important arrangement. The study of the braid arrangement was developed
from several ways. In this article, we prove that the module of differential operators on the braid arrangement is free by constructing

a basis. In addition, we discuss the action of the symmetric groupon the elements of the basis.

1. INTRODUCTION

The theory of hyperplane arrangements has been developed by many
researchers. The hyperplane arrangement defined by the direct productis so called the braid arrangement. The braid arrangement is the Coxeter arrangement of type
. It was proved by Saito [6] that Coxeter
arrangements are free. An excellent reference on arrangements is thebook by Orlik and Terao [5].
Let $K$ be a field, and $S=K[x_{1}, \ldots, x_{n}]$ be the polynomial ring ofvariables. Let $D^{(m)}(S)$be the module of differential
operators (of order m) of , whereis a multi-index. Fora central arrangement , we fix the defining polynomial $Q(\mathscr{A})=$
where $ker(p_{H})=H$ . We define the moduleof$m$-differential operators of order as follows:$A_{n-1}$$n$$:=\oplus_{|\alpha|=m}S\partial^{\alpha}$$S$$\alpha\in \mathbb{N}^{\ell}$$\mathscr{A}$$D^{(m)}(\mathscr{A})$$\prod_{H\in\ovalbox{\tt\small REJECT}}p_{H}$$\mathscr{A}$$D^{(m)}(\mathscr{A});=\{\theta\in D^{(m)}(S)|\theta(Q(\mathscr{A})S)\subseteq Q(\mathscr{A})S\}$.

In the case $m=1,$
is the module of -derivations.
Holm began to study
in his $PhD$ thesis. Holm proved
the idealizer of the ideal generated by the defining polynomial of acentral arrangement is the direct sum of the module of -differential
operators. We can describe the ring of differential operators of thecoordinate ring of a central arrangement.
Holm proved thatare free for all $m\geq 1$ whenis a 2dimensional central arrangement. Let
be a generic arrangement. It
was already known thatis not free if $n>3,$$m<$$|\mathscr{A}|-n+1$ , and is free if $n>3,$$m=|\mathscr{A}|-n+1[2].In$$D^{(1)}(\mathscr{A})$$\mathscr{A}$$D^{(m)}(\mathscr{A})$$\mathscr{A}$$D^{(m)}(\mathscr{A})$$\mathscr{A}$$\mathscr{A}$$D^{(m)}(\mathscr{A})$$|\mathscr{A}|>n,$$|\mathscr{A}|>n,$

136
NORIHIRO NAKASHIMA

addition, the author, Okuyama and Saito [4] proved that

$D^{(m)}(\mathscr{A})$is$m>|\mathscr{A}|-n+1$ .
free if $n>3,$
be the braid arrangement. In this article, we prove thatLet
is free by constructing a basis, and calculate an action of the.
symmetric group on$|\mathscr{A}|>n,$$\mathcal{A}$$D^{(2)}(\mathcal{A})$$D^{(2)}(\mathcal{A})$2. DETERMINANTS OF MATRlCES
Let $n\geq m>0$ . Let denote the th elementary symmetric poly, we
nomial in $m$ variables. For $n>m$ andbydefine row vectors$\ell-$$e_{\ell}$$\{i_{1}, \ldots, i_{m}\}\subseteq\{1, \ldots, n\}$$v_{i_{1},\ldots,i_{m}}$(2.1)$v_{i_{1},\ldots,i_{m}}=(\ldots,$ $e_{\ell_{1}}(x_{i_{1}}, \ldots, x_{i_{m}})\cdots e_{\ell_{n-m}}(x_{i_{1}}, \ldots, x_{i_{m}}),$

and anareth square matrix. Namely$(\begin{array}{l}nm\end{array})-$$v_{i_{1},\ldots,i_{m}}$$E_{m}(x_{1}, \ldots, x_{n})$$E_{m}(x_{1}, \ldots, x_{n})=(\begin{array}{lll} | v_{i_{1}} \cdots i_{m} | \end{array})$$\ldots)_{0\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m}$

as the matrix whose rows$(\{i_{1}, \ldots, i_{m}\}\subseteq\{1, \ldots, n\})$.

We agree $E_{n}(x_{1}, \ldots, x_{n})=(1)$ .
Let$\Delta=\triangle(x_{1}, \ldots, x_{n})=\prod_{1\leq i<j\leq n}(x_{i}-x_{j})$be the difference product.
Theorem 2.1.
$\det E_{m}(x_{1}, \ldots,x_{n})=c\triangle(\begin{array}{l}n- 2m- 1\end{array})$,

where $c=\pm 1$ .
To prove the theorem above, we first consider the degree of thisbe the total sum of degrees of polynomials in
determinant. Let
the set$a_{n,m}$(2.2)$\{e_{\ell_{1}}\cdots e_{\ell_{n-m}}|0\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m\}$Namely$a_{n,m}= \sum_{0\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m}\deg e_{\ell_{1}}\cdots e_{\ell_{n-m}}$

Proposition 2.2.
$a_{n,m}=(\begin{array}{l}n2\end{array})(\begin{array}{ll}n -2m -1\end{array})$

.

.

.

,

137
THE BRAID HYPERPLANE ARRANGEMENT

Proof. It is clear that$\deg(e_{\ell_{1}}\cdots e_{\ell_{k}})=P_{1}+\cdots+\ell_{k}$is equal to the set(2.3). Since the set (2.2)$\bigcup_{1\leq k\leq n-m}\{e_{\ell_{1}}\cdots e_{\ell_{k}}|1\leq\ell_{1}\leq\cdots\leq\ell_{k}\leq m\}$,it follows that$a_{n,m}= \sum_{0\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m}(\ell_{1}+\cdots+l_{n-m})$$= \sum_{1\leq k\leq n-m}\sum_{0\leq\ell_{1}\leq\cdots\leq\ell_{k}\leq m}(\ell_{1}+\cdots+\ell_{n-m})$.

Therefore$a_{n,m}$$= \sum_{1\leq k\leq n-1-m}\sum_{0\leq\ell_{1}\leq\cdots\leq\ell_{k}\leq m}(\ell_{1}+\cdots+\ell_{n-m})+\sum_{1\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m}(l_{1}+\cdots+l_{n-m})$$=a_{n-1,m}+ \sum_{1\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m}((\ell_{1}-1)+\cdots+(l_{n-m}-1))+(n-m)(\begin{array}{ll}n -1m -1\end{array})$$=a_{n-1,m}+ \sum_{0\leq\ell_{1}\leq\cdots\leq\ell_{n-m}\leq m-1}(l_{1}+\cdots+\ell_{n-m})+(n-m)(\begin{array}{l}n-1m-1\end{array})$$=a_{n-1,m}+a_{n-1,m-1}+(n-m)(\begin{array}{l}n-1m-1\end{array})$.

The assertion is completed by induction.

$\square$

To prove Theorem 2.1, we use the following two lemmas. We agree$\det E_{0}=1$ .
Lemma 2.3. Assume that $n>m$ . The$\det E_{m}(x_{1}, \ldots, x_{n})$ is equal to$coeff_{\grave{l}}c\iota ent$of$\det E_{m-1}$ $(x_{2}, \ldots , x_{n})\cdot\det E_{m}(x_{2}, \ldots, x_{n})$$x_{1}^{(n-1)(\begin{array}{l}n-2m-1\end{array})}$in.

Proof. After fundamental operations, we may assume that the upperrows containand the rightcolums are of indexes. . . $\leq\ell_{n-m}\leq m$ . Let be the upper rightsubmatrix of, i.e.,$(\begin{array}{l}n-1m-l\end{array})$$(\begin{array}{l}n-lm-1\end{array})$$x_{1}$$A$$\ell_{1}\leq$$1\leq$$(\begin{array}{l}n-lm-1\end{array})\cross(\begin{array}{l}n-1m-1\end{array})$$E_{m}(x_{1}, \ldots, x_{n})$$E_{m}(x_{1}, \ldots, x_{n})=\{\begin{array}{lll} * AE_{m}(x_{2} \cdots x_{n})*\end{array}\}$Since the -degree
for $1\leq l_{1}\leq\cdots\leq\ell_{n-m}\leq m$ and the$x_{1}$

is equal to $n-m$-degree of every component$\deg_{x_{1}}e_{\ell_{1}}\cdots e_{\ell_{n-m}}(x_{1}, x_{i_{2}}, \ldots, x_{i_{m}})$$x_{1}$.

138
NORIHIRO NAKASHIMA

which does not appear in$(n-1)(\begin{array}{l}n- 2m- 1\end{array})$$(n-m)(\begin{array}{l}n- 1m- 1\end{array})$$=x_{1}$$x_{1}$is less than $n-m_{7}$ the term containing
appears only in $\det A\cdot\det E_{m}(x_{2}, \ldots, x_{n})$ .
$A$

It remains to prove that the coefficient of
to $\det E_{m-1}(x_{2}, \ldots, x_{n})$ . Since$x_{1}^{(n-1)(\begin{array}{l}n-2m- 1\end{array})}$in$\det$ $A$

is equal$e_{\ell}(x_{1}, x_{i_{2}}, \ldots, x_{i_{m}})=e_{\ell-1}(x_{i_{2}}, \ldots, x_{i_{m}})x_{1}+e_{\ell}(x_{i_{2}}, \ldots, x_{i_{m}})$,we have that$e_{\ell_{1}}\cdots e_{\ell_{n-m}}(x_{1}, x_{i_{2}}, \ldots, x_{i_{m}})=e_{\ell_{1}-1}\cdots e_{l_{n-m}-1}(x_{i_{2}}, \ldots, x_{i_{m}})x_{1}^{n-m}+b$where $\deg_{x_{1}}b<n-m$ . Therefore we conclude that$\det A=\det E_{m-1}(x_{2}, \ldots , x_{n})x_{1}^{(n-m)(\begin{array}{l}n- 1m- 1\end{array})}+B$

where$\deg_{x}1B<(n-m)(\begin{array}{l}n-1m-1\end{array})=(n-1)(\begin{array}{l}n-2m-l\end{array})$.

$\square$In general, the following holds.

Lemma 2.4. Let
be a ring of chamcteristic zero. Let $f\in R[y]$ be. Suppose $\deg f\geq m$ . Then $(y-\alpha)^{m}$a nonzero polynomial anddivides if and only iffor $\ell=0,1,$ , $m-1$ .
$R$$\alpha\in R$$f^{(\ell)}(\alpha)=0$$f$$\ldots$

Proof of Theorem 2.1. We will prove the theorem by induction on.
It is clear that the case $n=m$ . Assume that $n>m$ . By renumberingdefined by (2.1), we nlay assume that -degrees of all$v_{1},$$\ldots,$$n$$x_{1}$$v_{(_{m}^{n})}$components of$v_{\ell}$

are

$0$

for

$(\begin{array}{l}n-lm-l\end{array})+1\leq\ell\leq(\begin{array}{l}nm\end{array})$. Thus we see that for$1\leq k\leq(\begin{array}{l}n-2m-l\end{array})-1$

(2.4)

where

$\det E_{m}(x_{1}, \ldots, x_{n})^{(k)}=\sum_{(_{m}^{n})}\det k_{1}+\cdots+k=k(\begin{array}{l}|v_{i}^{(k_{i})}|\end{array})$$f^{(k)}= \frac{\partial^{k}}{\partial x_{1}^{k}}(f)$$\frac{\partial}{\partial x_{1}}v_{j}=0$$0,$$\ldots,$for$k_{(_{m}^{n})}=0$

there exist

for any $f\in S$ and $v^{(k)}=(v_{1}^{(k)},$$(\begin{array}{l}n-1m-l\end{array})+1\leq j\leq(\begin{array}{l}nm\end{array})$$\ldots,$, we may consider$v_{\ell}^{(k)})$. Since$k(\begin{array}{l}n- 1m-1\end{array})+1=$

on (2.4). For any $\{i_{2}, \ldots, i_{m}\}\subseteq\{2, \ldots, t-1, t+1, \ldots, n\}$ ,$(\begin{array}{l}n-1m-1\end{array})+1\leq\ell\leq(\begin{array}{l}nm\end{array})$such that$v|_{x=x_{t}}=v_{\ell}$ .

A cardinality of $\{\{i_{2}, \ldots , i_{m}\}\subseteq\{2, \ldots, t-1, t+1, \ldots, n\}\}$ equals
We have that every term of RHS on (2.4) has $k_{i}=0$ withThus we conclude that$(\begin{array}{l}n-2m-1\end{array})$$1\leq i\leq(\begin{array}{l}n-lm-l\end{array})$$\det E_{m}(x_{1}, \ldots, x_{n})^{(k)}|_{x_{1}=x_{t}}=0$

.

.

139
THE BRAID HYPERPLANE ARRANGEMENT

for$1\leq k\leq(\begin{array}{l}n-2m-1\end{array})-1$(2.5). By Lemma 2.4,.

$\det E_{m}(x_{1}, \ldots, x_{n})\in\prod_{i\neq 1}(x_{1}-x_{i})S$

By the induction hypothesis and Lemma 2.3, we see that the coefficientof$x_{1}^{(n-1)(\begin{array}{l}n- 2m- 1\end{array})}$in $\det E_{m}(x_{1}, \ldots, x_{n})$ is$c\triangle(x_{2}, \ldots, x_{n})^{(\begin{array}{l}n- 3m- 2\end{array})}\cdot\triangle(x_{2}, \ldots, x_{n})^{(\begin{array}{l}n- 3m- 1\end{array})}=c\triangle(x_{2}, \ldots, x_{n})^{(\begin{array}{l}n- 2m- 1\end{array})}\neq 0$

where $c=\pm 1$ . So it follows form (2.5) that$\det E_{m}(x_{1}, \ldots , x_{n})\in\triangle(x_{1}, \ldots, x_{n})^{(\begin{array}{l}n- 2m- 1\end{array})}S\backslash \{0\}$.

By Proposition 2.2, the degree of $\det E_{m}(x_{1}, \ldots, x_{n})$ equals

Comparing degrees, we see thatover $c=\pm 1$ .

$(\begin{array}{l}n2\end{array})(\begin{array}{l}n-2m-l\end{array})$$\det E_{m}(x_{1}, \ldots, x_{n})=c\triangle(\begin{array}{l}n- 2m- 1\end{array})$

3. ELEMENTS OF THE MODULE OF.

. More$\square$-DIFFERENTIAL OPERATORSON THE BRAID ARRANGEMENT$\mathcal{A}$Throughout the remaining of this paper, let

be the n-th braid hyperplane arrangement, and let
be the module of A-differential
operators which preserve the ideal generated by. We assume thatthe characteristic of $K$ is zero. By [3, Proposition 2.3] and [3, Theorem2.4], we have$\mathcal{A}$$D^{(m)}(\mathcal{A})$$Q(\mathcal{A})$(3.1)$D^{(m)}( \mathcal{A})=\bigcap_{H\in \mathcal{A}}D^{(m)}(p_{H}S)$,

where $D^{(m)}(p_{H}S)=\{\theta\in D^{(m)}(S)|\theta(p_{H}x^{\alpha})\in p_{H}S$ for any $|\alpha|=m-1\}$
for
.
Put
$H\in \mathcal{A}$$h_{t}=(x_{t}-x_{1})\cdots(x_{t}-x_{t-1})(x_{t}-x_{t+1})\cdots(x_{t}-x_{n})$Proposition 3.1. Let
(3.2)

$k$

be a positive integer. The opemtors$\eta_{t,k}=h_{t}\sum_{|\alpha|=m;\alpha_{t}\geq k}\frac{1}{\alpha!}\partial^{\alpha}$belong to$D^{(m)}(\mathcal{A})$.

$(t=1, \ldots , n)$.

Proof. For any $1\leq i<j\leq n$ such that $i\neq t,j\neq t$ and$|\beta|=m-1$ ,$\sum_{|\alpha|=m;\alpha_{t}\geq k}\frac{1}{\alpha!}\partial^{\alpha}((x_{i}-x_{j})x^{\beta})=\{\begin{array}{ll}0 if \beta_{t}<k1-1=0 if \beta_{t}\geq k.\end{array}$$\beta$with

140
NORIHIRO NAKASHIMA

for any
This implies that$i<j\leq n$ . It follows from (3.1) that operators (3.2) belong to
as required.
$h_{t} \sum_{|\alpha|=m;\alpha_{t}\geq k}\frac{1}{\alpha!}\partial^{\alpha}\in D^{(m)}((x_{i}-x_{j})S)$$1\leq$$D^{(m)}(\mathcal{A})\square$

We consider other polynomials relating with elementary symmetricbe the th elementary symmetricpolynomials. Let
polynomial in variables. For a multi-index , we define$e_{\ell}^{k}=e_{\ell}(y_{1}, \ldots, y_{k})$$k$$\ell-$$\alpha$$e_{\ell}^{k}(\alpha)=e_{\ell}^{k}(x_{1}, \ldots , x_{1}, x_{2}, \ldots, x_{2}, \ldots , x_{n}, \ldots , x_{n})\in S$

where the number of$x_{t}$

Proposition 3.2. Let
nonnegative integers

is$k$$\ell_{1},$(3.3)$\alpha_{t}$.

be a positive integer. For any sequence of, , the opemtor$\ell_{k}$$\ldots$$\theta_{\ell_{1},\ldots,\ell_{k}}=\sum_{|\alpha|=m}e_{p_{1}}^{m}(\alpha)\cdots e_{\ell_{k}}^{m}(\alpha)\frac{1}{\alpha!}\partial^{\alpha}$belongs to$D^{(m)}(\mathcal{A})$Proof. Since.

.

$e_{\ell}(y_{1}, \ldots, y_{m})=e_{l}(y_{1}, \ldots, y_{m-1})+e_{\ell-1}(y_{1}, \ldots, y_{m-1})y_{m}$for $1\leq i<j\leq n$ and$\beta$with $|\beta|=m-1$ ,$\theta_{\ell_{1},\ldots,\ell_{k}}((x_{i}-x_{j})x^{\beta})$$=e_{\ell_{1}}^{m}(\beta+e_{i})\cdots e_{p_{k}}^{m}(\beta+e_{i})-e_{\ell_{1}}^{m}(\beta+e_{j})\cdots e_{\ell_{k}}^{m}(\beta+e_{j})$$=(e_{\ell_{1}}^{m}(\beta)+e_{p1}^{m}(\beta)x_{i})\cdots(e_{\ell_{k}}^{m}(\beta)+e_{\ell_{k}}^{m}(\beta)x_{i})$$-(e_{p_{1}}^{m}(\beta)+e_{\ell_{1}}^{m}(\beta)x_{j})\cdots(e_{p_{k}}^{m}(\beta)+e_{\ell_{k}}^{m}(\beta)x_{j})$Substitutefor , then we getfollows that a polynomial
divides a polynomial. Therefore we concludesoby (3.1).
$x_{j}$.
. This$\theta_{\ell_{1},\ldots,p_{k}}((x_{i}-x_{j})x^{\beta})|_{x_{i}=x_{j}}=0$$x_{i}$$\theta_{\ell_{1},\ldots,p_{k}}((x_{i}-x_{j})x^{\beta})$$x_{i}-x_{j}$$\theta_{\ell_{1},\ldots,\ell_{k}}\in D^{(m)}((x_{i}-x_{j})S)$

4. A BASIS FOR$D^{(2)}(\mathcal{A})$$\theta_{\ell_{1},\ldots,\ell_{k}}\in D^{(m)}(\mathcal{A})$$\square$

AND ITS REPRESENTATIONIn this section, we assume that the characteristic of $K$ is zero and$K$ is a algebraically closed.
We find a basis for the module
relating with the Specht modules. For $f,$ $g\in S$ , it is convenient to.
write $f=g$ if $f=cg$ for some$D^{(2)}(\mathcal{A})$$c\in K\backslash \{0\}$Theorem 4.1. Let(4.1)$\eta_{t}=\eta_{t,1}$. The set$\{\eta_{1}, \ldots, \eta_{n}\}\cup\{\theta_{\ell_{1},\ldots,\ell_{n-2}}|0\leq\ell_{1}\leq\cdots\leq\ell_{n-2}\leq 2\}$

forms a basis for$D^{(2)}(\mathcal{A})$

.

,

141
THE BRAID HYPERPLANE ARRANGEMENT

Proof. We have already seen, by Proposition 3.1 and Proposition 3.2,
and.
By Saito-Holm criterion [8, Theorem 4.10.], it is sufficient to showthat$\eta_{t,1}\in D^{(2)}(\mathcal{A})$$\theta_{\ell_{1},\ldots,\ell_{k}}\in D^{(2)}(\mathcal{A})$(4.2)$\det M_{m}(\eta_{t}, \theta_{\ell_{1},\ldots,\ell_{n-2}}|t=1, \ldots, n, 0\leq\ell_{1}\leq\ldots\leq p_{n-2}\leq 2)=Q(\mathcal{A})^{n}$where
coefficient matrix of the operators in (4.1). By Theorem 2.1.

$M_{m}(\eta_{t}, \theta_{\ell_{1},\ldots,\ell_{n-2}}|t=1, \ldots, n, 0\leq\ell_{1}\leq\cdots\leq\ell_{n-2}\leq 2)$$*$$\det M_{m}=Q^{2}I_{n}0$$=Q^{2}\cdot Q^{n-2}=Q^{n}$$E_{m}(x_{1}, \ldots, x_{n})$is the,as required.

$\square$Define the K-vector space$V= \sum_{t=1}^{n}K\eta_{t}+\sum_{0\leq\ell_{1}\leq\cdots\leq\ell_{n-2}\leq 2}K\theta_{\ell_{1},\ldots,\ell_{n-2}}$.

Now we retake a basis which is also a basis for the decomposition of
into Specht modules.
Letbe a Young diagram ofcells. We say that is a partition of and write. Define Tab(resp. STab ) be the set of Young tableaux (resp. standard tableaux)of shape .
We say that $T(i, j)$ is a number in the $(i,j)$ -box (i.e. the box of i-th
row and j-th column) of a tableau $T\in$ Tab . Let$V$$\lambda=(\lambda_{1}, \ldots, \lambda_{n})(\lambda_{1}\geq\cdots\geq\lambda_{n}\geq 0)$$\lambda$$n$$\lambda\vdash n$$n$$(\lambda)$$(\lambda)$$\lambda$$(\lambda)$$\triangle\tau=\prod_{j=1}^{\lambda_{1}}\prod_{1\leq i_{1}<i_{2}\leq\lambda_{j}’}(x_{T(i_{1},j)}-x_{T(i_{2},j)})\in S$

be the Specht polynomial, for $T\in$ Tab
an $K[S_{n}]$ -module$(\lambda)$. For each partition$\lambda$

of ,
$n$

$V_{\lambda}= \sum_{T\in Tab(\lambda)}K\triangle\tau$is called the Specht module. The following proposition is well known(cf. [1]).

Proposition 4.2. Let.
(1) The set
STabforms a K-basis for .
(2) The representation
is irreducible. Every irreducible representation is isomorphic to one of .
$\lambda\vdash n$$\{\triangle\tau|T\in$$(\lambda)\}$$V_{\lambda}$$V_{\lambda}$$V_{\lambda}$

142
NORIHIRO NAKASHIMA

The symmetric group$S_{n}$

acts on the Weyl algebra by$\sigma\cdot x_{i}=x_{\sigma^{-1}(i)}$.
$D^{(m)}(S)$ andfor$\sigma\in S_{n}$,$\sigma\cdot\partial_{i}=\partial_{\sigma^{-1}(i)}$Then for any homogeneous differential operator$\sigma\in S_{n}$(4.3)$\theta\in$,$(\sigma\theta)(f)=\sigma(\theta(\sigma^{-1}f))$$(f\in S)$

We show that the symmetric group also acts on.

$D^{(m)}(\mathcal{A})$;

Proposition 4.3.
$S_{n}.D^{(m)}(\mathcal{A})\subseteq D^{(m)}(\mathcal{A})$Proof. Let$\theta\in D^{(m)}(\mathcal{A})$and$\sigma\in S_{n}$.

. For $f\in S$ , there exist $g\in S$ suchthat$\theta(sgn(\sigma^{-1})Q(\sigma^{-1}f))=Qg$.

We see that$(\sigma\theta)(Qf)=\sigma(\theta(\sigma^{-1}(Qf)))=\sigma(\theta(sgn(\sigma^{-1})Q\sigma^{-1}(f)))$$=\sigma(Qg)=sgn(\sigma)Q\cdot\sigma g\in QS$

by (4.3). Therefore$\sigma\theta\in D^{(m)}(\mathcal{A})$.

$\square$. It is clear thatLet
. So we see thatforis isomorphic to $V_{(n)}=K$ .
It only remains to decompose
into Specht modules.
For $i<j$ , a transposition $(i, j)$ acts on ’s as follows:$\sigma\theta=\theta$$\theta\in\{\theta_{\ell_{1},\ldots,\ell_{n-2}}|0\leq\ell_{1}\leq\cdots\leq\ell_{n-2}\leq 2\}$
$K\theta$$\sigma\in S_{n}$$W= \sum_{t=1}^{n}K\eta_{t}$$\eta$$(i,j)\eta_{i}=\eta_{j},$ $(i,j)\eta_{j}=\eta_{i},$ $(i,j)\eta_{k}=\eta_{k}$$(k\neq i,j)$.

We have that $W$ is isomorphic to the $K[S_{n}]$ -module of homogeneouspolynomials of degree 1. So it is well-known a $K[S_{n}]$ -module decomposition$W=K( \eta_{1}+\cdots+\eta_{n})\oplus\sum_{t=2}^{n}K(\eta_{1}-\eta_{t})\simeq V_{(n)}\oplus V_{(n-1,1)}$

We retake a basis for for$D^{(2)}(\mathcal{A})$.

.

Corollary 4.4. The set$\{\eta_{1}+\cdots+\eta_{n}, \eta_{1}-\eta_{2}, \ldots, \eta_{1}-\eta_{n}\}\cup\{\theta_{\ell_{1},\ldots,\ell_{n-2}}|0\leq\ell_{1}\leq\cdots\leq\ell_{n-2}\leq 2\}$

forms a basis for$D^{(2)}(\mathcal{A})$.

THE BRAID HYPERPLANE ARRANGEMENTREFERENCES[1] W. Fulton, Young Tableaux. Cambridge University Press, 1997.
[2] P. Holm, Differential Opemtors on Arrangements of Hyperplanes. PhD. Thesis,
Stockholm University, (2002).
[3] P. Holm, Differential Opemtors on Hyperplane Armngements. Comm. Algebra
32 (2004), no.6, 2177-2201.
[4] N. Nakashima, G. Okuyama and M. Saito, The Freeness and Minimal Free
Resolutions of Modules of Differential Opemtors of a Generic Hyperplane Armngement. J. Algebra, 351 (2012), 294-318.
[5] P. Orlik and H. Terao, Awangements of Hyperplanes. Grundlehren dermatematischen Wissenschaften 300, Springer-Verlag, 1992.
[6] K. Saito, On the uniformization of complements of discriminantal loci. in Seminar Notes. Amer. Math. Soc. Summer Institute, Williamstown, MA, (1975).
[7] K. Saito, Theory of loganthmic differential forms and logarithmic vector fields.
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265-291.
[8] J. Snellman, A Conjecture on Pioncar\’e-Betti Series of Modules of Differential
Opemtors on a Generic Hyperplane Arrangement. Experiment. Math. 14 (2005),
no.4, 445-456.
E-mail address: naka-n@math.sci.hokudai.ac.jp

DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, HOKKAIDO
UNIVERSITY, SAPPORO, 060-0810, JAPAN143