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\newtheorem{theo}{�藝}[section]
\newtheorem{df}[theo]{��`}
\newtheorem{lemma}[theo]{���}
\newtheorem{pr}[theo]{����}
\newtheorem{col}[theo]{�n}
\newtheorem{ch}[theo]{����}
\newtheorem{ex}[theo]{��}
\newtheorem{fact}[theo]{����}
\newtheorem{strategy}[theo]{����}

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\title{�Q�P���T�ώ��x�N�g����Ԃ̑��Εs�ώ���\\
\hfill �\���ɂ���}
\author{
���ؑ]\ �x�`\ \ �{��\ �� \\
�}�g��w\ ���w�n}
\date{}



\begin{document}

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\maketitle


\section{�͂��߂�}
�{�����́A$2$ �P���T�ώ��x�N�g����Ԃ̑��Εs�ώ��̋�̓I�ȍ\�����s�Ȃ����̂ł���B$1961$ �N�������v���ɂ�薼�t����ꂽ�u�T�ώ��x�N�g����ԁv�́A���ޗ��_�⑊�Εs�ώ��̍\���A�T�ώ��x�N�g����Ԃ̃[�[�^�֐��ȂǑ���ɂ킽�茤�����i�߂��Ă����B�{�_���̎�|�ł��鑊�Εs�ώ��̋�̓I�ȍ\���ɂ��Ă̌������A����\���ɂ����ẮA�������v���A�ؑ��B�Y���ɂ�肢�����������Č��肳��A�c��̋�Ԃɑ΂��Ă͍s�Җ��F���A�����[�V���ɂ���Č��肳��Ă���B�܂��A����łȂ��ꍇ�Ɋւ��ẮA�P���T�ώ��x�N�g����Ԃɑ΂��Ėؑ����ɂ�茈�肳��Ă���B�����ŁA�{�����͂��̌p���Ƃ��� $2$ �P���T�ώ��x�N�g����Ԃ̑��Εs�ώ�����̓I�ɍ\�����悤�Ƃ������݂Ŏn�܂����B

[K-K-I-Y] �̒��ŁA$type\ 1$ �� $2$ �P���T�ώ��x�N�g����Ԃ����ނ���Ă���A$46$ �ɕ��ނ��ꂽ {\bf �����Q�P���T�ώ��x�N�g�����}  �� $26$ �ɕ��ނ��ꂽ {\bf �񐳑��Q�P���T�ώ��x�N�g�����} �ɂ��Ċe�X�̑��Εs�ώ��̋�̓I�ȍ\�����s�Ȃ����B

���Εs�ώ��̍\���Ɋւ��āA�������̍\���@�ȂǊ��m�̍\���@�����p����Ɠ����ɁA�I���W�i���ȍ\�����s�Ȃ����B���ɁA$reg.8$ �� $reg.40$�Ȃǂɂ����āA $Mathematica$ ��p�����v�Z���@�́A���ړI�ɑ��Εs�ώ������߂�P�̎�i�ł���A���� $type\ 1$ �� $2$ �P���T�ώ��x�N�g����ԑS�Ă̏ꍇ�̍\�����������Ă���A�_�������M���ł���B

�Ȃ��{���e�̓��e�́A�ؑ��B�Y�A���ؑ]�x�`�A���т݂䂫�A�{�����̋��������̈�[�ł���B


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\section{�T�ώ��x�N�g����Ԃ̒�`�Ƒ��Εs�ώ��̊�{�I����}
�ȉ��A�͕̂��f���� $\mathbb{C}$ �ōl����B

\begin{df}[�T�ώ��x�N�g�����]\label{}
$\newline$
\hspace{1cm}$\begin{array}{ccl}
G &:& �A�����`�㐔�Q \\
V &:& \mathbb{C} ��̗L�����x�N�g����� \\
\rho &:& G\longrightarrow GL(V)~:~ �L���\�� 
\end{array}$ \\
�Ƃ���B���̂Ƃ��A\\
\hspace{1cm}${}^\exists  \mbox{x}_{\rm 0}\in V~s.t.~\overline{\rho(G)\cdot \mbox{x}_{\rm 0}} =V$~~�i�A���A $\overline{{{\hspace{.7cm}}^{}}^{}}$ �� $Zariski$ ��j\\
�ł���Ƃ��A���Ȃ킿�L���\�� $\rho$ �� $Zariski$ �ʑ����f���� $G$�|�O�������Ƃ��A $3$ �g $(G,\rho,V)$ ���T�ώ��x�N�g����� $(prehomogeneous~vector~space)$ �Ƃ�ԁB\\
\end{df}


\begin{df}[���Εs�ώ�]\label{}
$\newline$
�T�ώ��x�N�g����� $(G,\rho,V)$ �ɂ��āA $V$ ��̍P���I�ɂ͗�ł͂Ȃ��L���֐� $f(\mbox{x})$ �����Εs�ώ� $(relative~invariant)$ �ł���Ƃ́A
\[ {}^\exists\chi~:~ �L���w�W ~s.t.~f(\rho(g)\mbox{x})=\chi(g)f(\mbox{x})~~(\mbox{x}\in V-S,g\in G) \]
�����藧���Ƃł���B���̂Ƃ��A$f\longleftrightarrow \chi$ �ƋL���B
\end{df}

\begin{theo}\label{}
$\newline$
$(G,\rho,V)~:~P.V.~s.t.~~V-S=\rho(G)\cdot \mbox{x}_{\rm 0}$ �ɂ��āA$S$ �́A$S=S_1\cup\cdots\cup S_r\cup S'$ �Ƃ�����B�A���A$S_i=\{ \mbox{x}\in V~|~f_i(\mbox{x})=0 \}~(1\le i\le r)$ �� $S$ �̗]���� $1$ �̊��񐬕��ł���A$S'$ �͗]���� $2$ �ȏ�̕����W���ł���B���̂Ƃ��A���񑽍��� $f_1,\cdots,f_r$ �͑㐔�Ɨ��ȑ��Εs�ώ��ŁA�C�ӂ̑��Εs�ώ� $f(x)$ ��
\[ f(\mbox{x})=cf_1(\mbox{x})^{m_1}\cdots f_r(\mbox{x})^{m_r}~~~~(c\in\mathbb{C}~,~(m_1,\cdots,m_r)\in \mathbb{Z}^r)\]
�ƈ�ӓI�ɂ�����B
\end{theo}

���̈Ӗ��� $f_1,\cdots,f_r$ �� $P.V.~~(G,\rho,V)$ �̊�{���Εs�ώ��ł���Ƃ����B\\
�ȉ��A���̑��̐�����񋓂���B

\begin{itemize}
\item $\begin{array}{ccc}
f_1&& \\
&\rotatebox[origin=c]{315}{$\longleftrightarrow$} \\
&&\chi~~\Longrightarrow {}^\exists c~:$ �萔 $~s.t.~~f_2=cf_1\\
&\rotatebox[origin=c]{45}{$\longleftrightarrow$} \\
f_2&&
\end{array}$ 
\item ���Εs�ώ��͐Ď����ł���B
\item $X(G)=\{\chi~:~G\longrightarrow GL_1~|~�L���w�W\}$ �̕����Q�A $X_1(G)$ �� \\
\hspace{1cm}$X_1(G)=\{\chi\in X(G)~|~f\longleftrightarrow\chi$ �ƂȂ鑊�Εs�ώ� $f$ �����݂��� $\}$ \\
�Ƃ��A$v_0$ �𐶐��_�A$G_{v_0}=\{g\in G~|~\rho(g)v_0=v_0\}$ �����̓��������Q�Ƃ���B���̂Ƃ��A\\
\hspace{1cm} $X_1(G)=\{\chi\in X(G)~|~\chi |_{G_{v_0}}=1\}$ \\
�Ɠ����Â�����B
\item �����I���������Q $G_{v_0}$ �ƌ����q�Q $[G,G]$ �Ő�������� $G$ �̐��K�����Q $G_1$ �͐����_�̂Ƃ���Ɉ˂炸���܂�A $G/G_1$ �̎w�W�Q $X(G/G_1)$ �́A
\[ X_1(G)=\{ \chi\in X(G)~|~f\leftrightarrow \chi �ƂȂ鑊�Εs�ώ� f �����݂���B\} \]
�Ɠ��ꎋ�ł���B
\item $X_1(G)$ �͗L���������R $abel$ �Q�ɂȂ�A���̊K�� $N=rank~X_1(G)$ ����{���Εs�ώ��̌��ł���B
\end{itemize}

��X�͑��Εs�ώ��̍\���ɂ��āA�ȉ��̌����Ɋ�Â����B

\begin{strategy}\label{}
$\newline$
$(G,\rho,V)\longrightarrow (G',\rho',V')$ �� $equivariant~~map$ �Ƃ́A\\
\hspace{1cm}$\left\{\begin{array}{l}
V\stackrel{\varphi}{\longrightarrow} V'~~:~~generically~~surjective~~map \hspace{1cm}(\Longleftrightarrow~\overline{\varphi(V)}=V') \\
G\stackrel{\psi}{\longrightarrow} G'~~:~~\hom
\end{array}\right. \\
\hspace{1cm}s.t.~~\varphi(\rho(g)v)=\rho'(\psi(g))\phi(v)~~~\left({}^\forall g\in G~,{}^\forall v\in V\right)$
\end{strategy}

���̌���������̏ꍇ�ɓ��Ă͂߂�ƗႦ�΁A�^����ꂽ�T�ώ��x�N�g����� $(G_1\times G_2,\rho_1\otimes \rho_2,V_1\otimes V_2)$ �ɑ΂��� $equivariant$ �� $(G_1,\rho',W)$ �ւ� $generically$\ $surjective$\ $map$ ���l����Ƃ������Ƃł���B���̑S�˂̍����Ƀ|�C���g������A�{�����ōł���J�����_�ł���B���̃A�C�f�A�͍���̌����i�R�P���T�ώ��x�N�g����Ԃ� $type$ �U�̂Q�P���T�ώ��x�N�g����Ԃ̑��Εs�ώ��̋�̓I�\��)�ɖ𗧂��̂Ɗm�M����B



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\section{���Εs�ώ��̋�̓I�\��}


���Εs�ώ��̋�̓I�\���ɂ�����p�������@�Ƃ��āA�����̍\���@�͖ܘ_�p���������ɗp��������������ɋ�����B\\
{\bf{\underline{1.\ ���� map}}}\footnote{�_�� [O1] �ɂ����ė����������������ʑ��ŁA�����ł�  $\beta$ �������Ă΂��Ă����������B}

����̌��s�� $X\in Alt_{2n+1}$ �ɑ΂��āA��p�� $X\longmapsto AX{}^tA~~(A\in GL_{2n+1})$ �Ƃ���B�Ⴆ�� $n=2$ �̂Ƃ��A

\begin{eqnarray*}
\begin{array}{ccc}
\beta~:~ Alt_5\times Alt_5 & \longrightarrow&  \mathbb{C}^5 \\
~~~(X,Y) &\longmapsto& \left(\begin{array}{c}\beta_1(X,Y)\\\vdots\\\beta_5(X,Y)\end{array}\right)
\end{array}
\end{eqnarray*}
���l����B�����ŁA
\[ \beta_i(X,Y)=(-1)^{i+1}\left\{Pf((X+Y)^{(i)})-Pf((X)^{(i)})-Pf((Y)^{(i)})\right\} \]
�ł���B�A���A$X^{(i)}$ �� $X$ ���� $i$ �s $i$ ��������� $4$ ���̌��s��ł���B\\
���� $\beta$ �̓p�t�B�A���̒�`���A�Ώ̑o���`�ł��邱�Ƃ�������B�܂��A$A\in GL_5$ �̍�p�ŁA
\[ \beta(X,Y)\longmapsto (\det A)~{}^tA{}^{-1}\beta(X,Y) \]
�����藧�B���� $\beta$ �𗎍� map �ƌĂԁB���̎ʑ��͋������̌��s��ɑ΂��Ă��قړ��l�ɒ�`�����B�{�_���ő��Εs�ώ����\������ɂ�����A�����p�����B\\
{\bf{\underline{2.\ �R�p�t�B�A��}}}

�p�t�B�A���͔����������̃p�b�t�`���ŗL���ȃp�b�t $(\mbox{J}.F.P \! faf\! f)$ �ɂ���āA�񏥂��ꂽ���̂ł���B�����ł́A�s�񎮂ɂ��p�t�B�A�����`����B
$A=(a_{ij})\in Alt_{2n}$ �Ƃ���B�����ŁA$Alt_{2n}=\{x\in M_{2n}~|~X=-{}^tX\}$ �ł���A���s��Ƃ�΂��B

\begin{df}[�p�t�B�A��]
$\newline$ $X\in Alt_{2n}$ �ɑ΂��āA�ȉ��̏��������� $n$ �������� $Pf(X)$ ���p�t�B�A�� $(P\! faf\! fian)$ �Ƃ����B
\begin{eqnarray*}
\left \{ \begin{array}{l}
Pf(X)^2=\det ~X \\
Pf(\left(\begin{array}{c|c}
    O& I_n \\ \hline
    -I_n& O
\end{array}\right) )=(-1)^{\frac{n(n-1)}{2}}
\end{array}\right .
\end{eqnarray*}
���� $2n$ ���̃p�t�B�A�� $Pf(X)$ �� $(1,2,\cdots,2n)$ �ƕ\�L���邱�Ƃ�����B
\end{df}
���ɁA�������藧�B
\begin{pr}
$X\in Alt_{2n}$ �ɑ΂��Ď������藧�B
\[ Pf(AX{}^tA)=\det A\cdot Pf(X) ~~~(A\in GL_{2n})\]
\end{pr}

�s��̏��������l�@����Ƃ��A�]���q�s���p���邱�Ƃ����X����B�]���q�s��͍s�񎮂�p���Ē�`�����킯�����A�����ł�����p�t�B�A���Ɋg�����āA�]���q�s��ɑΉ�����s����l���邱�Ƃ͂������R�ł���B�K���ȓ��{��󂪌�������Ȃ��̂ŁA���ڃR�p�t�B�A���ƌĂԂ��Ƃɂ���B�܂��A�����Œ��ӂ��邱�ƂɁA�p�t�B�A���͊���̌��s��ɑ΂��čP���I�� $0$ �ɂȂ邱�Ƃł���B���Ȃ킿�A�R�p�t�B�A���͋��������s��ɂ��Ă̂ݒ�`�����B

\begin{df}[�R�p�t�B�A��]
$\newline X\in Alt_{2n}$ �ɑ΂��āA
\[ X\Delta(X)=-Pf(X)I_{2n} \]
�Œ�߂� $\Delta(X)\in Alt_{2n}$ �� $X$ �̃R�p�t�B�A���ƌĂԁB
\end{df}
\begin{pr}
$\newline A\in GL_{2n}$ �̂Ƃ��A
\[ \Delta(AX{}^tA)=\det A~{}^tA^{-1}\Delta(X)A^{-1} \]
\end{pr}

\noindent{\bf{\underline{3.\ �]���q�s��}}}

�R�p�t�B�A���͋������̌��s��ɂ��Ē�`����邪�A�����Ŋ���̌��s��Ɋւ��ėގ����l����B���Ȃ킿�A����̌��s��ɑ΂���]���q�s������̂悤�ɒ�`����B

\begin{df}[�]���q�s��]
$\newline X\in Alt_{2n+1}$ �ɑ΂��āA�]���q�s�� $\Delta(X)=(x'_{ij})\in Sym_{2n+1}$ �����̂悤�ɒ�`����B
\[ x'_{ij}=(-1)^{i+j}~\det X^{(i;j)} ~~~(1\le i<j\le 2n+1) \]
�����ŁA$X^{(i;j)}$ �� $X$ ���� $i$ �s�� $j$ ��������� $2n\times 2n$ �s��ł���B
\end{df}
\begin{pr}
$\newline A\in GL_{2n+1}$ �̂Ƃ��A
\[ \Delta(AX{}^tA)=(\det A)^2~{}^tA^{-1}~\Delta(X)~A^{-1} \]
\end{pr}

\noindent{\bf{\underline{4.\ �R���s���[�^ $(Mathematica)$ ��p�����\�����@}}}






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\section{2-Simple~P.V.�̑��Εs�ώ��̍\��}

�ȉ��A����̂Q�|�P���T�ώ��x�N�g����Ԃɑ΂��āA���Εs�ώ��̋�̓I�\�����Љ��B\\

\begin{itemize}
\item $reg.1~~(GL_1^2\times SL_4\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda_1\otimes\Lambda_1~,~Alt_4\oplus Alt_4\oplus M(4,2))~~N=2$
\item $nonreg.13(b)~~(GL_1^2\times SL_7\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda^*_1\otimes 1~,~Alt_7\oplus Alt_7\oplus V(7)^*)~~N=1$
\item $nonreg.13(a)~~(GL_1^2\times SL_7\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda_1\otimes 1~,~Alt_7\oplus Alt_7\oplus V(7))~~N=1$
\item $reg.8~~(GL_1^2\times SL_5\times SL_8~,~\Lambda_2\otimes \Lambda_1+1\otimes\Lambda^*_1~,~\overbrace{Alt_5\oplus\cdots\oplus Alt_5}^8\oplus V(8)^*)~~N=2$
\end{itemize}



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\subsection{reg.1}

${\bf (1)~~(GL_1^2\times SL_4\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda_1\otimes\Lambda_1~,~Alt_4\oplus Alt_4\oplus M(4,2))~~N=2}$ \\

�Q�́A$GL_1\times SL_4\times GL_2$ �Ƃ��Ă悢�B$(\alpha;A,B)\in GL_1\times SL_4\times GL_2$ �ɑ΂��č�p�́A$(X,Y,Z)\longmapsto ((AX{}^tA,AY{}^tA){}^tB,\alpha~AZ{}^tB)$ �ł���B
\begin{itemize}
\item $1$ �ڂ��l����B
$X,Y\in Alt_4$ �ɂ��āA
\[ \beta(X,Y)=\{Pf((X+Y))-Pf(X)-Pf(Y)\} \]
�Ƃ����A
\[ \Phi(X,Y)=\left(\begin{array}{cc}
\beta(X,X)&\beta(X,Y) \\
\beta(X,Y)&\beta(Y,Y)
\end{array}\right)\in Sym_2 \]
�Ƃ����B���̂Ƃ��A�ȒP�Ȍv�Z�ɂ��A
 \[ \Phi((X,Y))\longmapsto B\Phi(X,Y){}^tB \]
�䂦�ɁA
\[ \det\Phi(X,Y)\longmapsto \det\Phi((AX{}^tA,AY{}^tA){}^tB)=(\det B)^2\det\Phi(X,Y) \]
����āA$1$ �ڂ̑��Εs�ώ��́A
\[ F_1(X,Y,Z)=\det\Phi(X,Y)\longleftrightarrow (\det B)^2 \]

\item $2$ �ڂ��l����B\\
$Z=\left(\begin{array}{c|c}z_{11}&z_{12}\\z_{21}&z_{22}\\z_{31}&z_{32}\\z_{41}&z_{42}\end{array}\right)\in M(4,2)$ �ɑ΂��āA\\
$F_{ij}(Z)=\det\left(\begin{array}{cc}z_{i1}&z_{i2}\\z_{j1}&z_{j2}\end{array}\right)~~~(i<j)$ �Ƃ����A$A=(a_{ij})\in GL_4~,~B=(b_{ij})\in GL_2$ �Ƃ����B\\
���̂Ƃ��A
\[ F_{ij}(Z{}^tB)=\det B~F_{ij}(Z) \]
�����藧���Ƃɒ��ӂ���B�����ŁA\\
$F(Z)=\left(\begin{array}{cccc}0&F_{12}(Z)&F_{13}(Z)&F_{14}(Z)\\-F_{12}(Z)&0&F_{23}(Z)&F_{24}(Z)\\-F_{13}(Z)&-F_{23}(Z)&0&F_{34}(Z)\\-F_{14}(Z)&-F_{24}(Z)&-F_{34}(Z)&0\end{array}\right)\in Alt_4$ �Ƃ����B�ȒP�Ȍv�Z�ɂ��A
\[ F(Z)\longmapsto \alpha^2\det B~AF(Z){}^tA \]
��������B�����ŁA$\beta(X,Y)=\{Pf((X+Y))-Pf(X)-Pf(Y)\}$ �Ƃ����A
\[ \Omega(X,Y,Z)=\left(\begin{array}{cc|c}
\beta(X,X)&\beta(X,Y)&\beta(X,F(Z)) \\
\beta(X,Y)&\beta(Y,Y)&\beta(Y,F(Z)) \\\hline
\beta(X,F(Z))&\beta(Y,F(Z))&0
\end{array}\right)\in Sym_3 \]
�Ƃ����B�����ŁA$Pf(F(Z))=0$ ���A$\beta(F(Z),F(Z))=0$ �ł���B���� $\Omega(X,Y,Z)$ �͊ȒP�Ȍv�Z�ɂ��A�X�Ɏ������藧�B
\[ \Omega((AX{}^tA,AY{}^tA){}^tB,Z{}^tB)=B'\det\Omega(X,Y,Z){}^tB' ~~~(B'=\left(\begin{array}{c|c}{}^tB&\\\hline &\alpha^2\det B\end{array}\right)) \]
�䂦�ɁA
\[ \det\Omega(X,Y,Z)\longmapsto \det\Omega((AX{}^tA,AY{}^tA){}^tB,Z{}^tB)=\alpha^4(\det B)^4\det\Omega(X,Y,Z) \]
����āA$2$ �ڂ̑��Εs�ώ��́A
\[ F_2(X,Y,Z)=\det\Omega(X,Y,Z)\longleftrightarrow \alpha^4(\det B)^4 \]
\end{itemize}

�ȏ���A\\
\begin{eqnarray*}
\begin{array}{|c|}\hline
(1)~~(GL_1^2\times SL_4\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda_1\otimes\Lambda_1~,~Alt_4\oplus Alt_4\oplus M(4,2)) \\
\begin{array}{lllll}
F_1(X,Y,Z) &=& \det\Phi(X,Y) &\longleftrightarrow& (\det B)^2 \\
F_2(X,Y,Z) &=& \det\Omega(X,Y,Z) &\longleftrightarrow& \alpha^4(\det B)^4
\end{array} \\ \hline
\end{array}
\end{eqnarray*}


$\diamondsuit$ {\bf �����񐫁�} \\
�����_ $(X_0,Y_0,Z_0)=(e_1\land e_2~,~e_3\land e_4~,~(e_1+e_3,e_2+e_4))$ �ɂ����铙�������́A$\mathfrak{g}_{X_0,Y_0,Z_0}=\left\{ 0~,~0\right\}$ �ł���B\\
$\chi |_{G_{x_0}}=1$ ���A���������Q $G_{x_0}$ �̓K���Ȍ���I�сA���񐫂��`�F�b�N����B\\
\hspace{1cm}$\left\{ (-i~,~\left(\begin{array}{cc|cc}&&&i\\&&i&\\\hline &i&&\\i&&&\end{array}\right)
~,~\left(\begin{array}{cc}&1\\1&\end{array}\right) )\right\}\in G_{x_0}$\\
 ���l����B���̂Ƃ��A$\det B=\det \left(\begin{array}{cc}&1\\1&\end{array}\right)=-1$ ���A$\det B$ �ɑΉ����鑊�Εs�ώ��͑��݂��Ȃ��B�䂦�ɁA$F_1(X,Y,Z)$ �͊���ł���B\\
���ɁA$\alpha^2(\det B)^2=-1$ ���A$F_2(X,Y,Z)=G(X,Y,Z)^2$ �Ȃ�\���͖����B�����ŁA$F_2(X,Y,Z)=G(X,Y,Z)F_1(X,Y,Z)$ �ł���Ɖ��肷��B\\
���Εs�ώ� $F_1(X,Y,Z)$ �� $F_2(X,Y,Z)$ �̊e $X,Y,Z$ �Ɋւ��鎟���́A\\
\hspace{2cm}$\begin{array}{l}
\deg F_1=(2,2,0) \\
\deg F_2=(2,2,4)
\end{array}$ \\
����āA$G(X,Y,Z)$ �̎����́A$(0,0,4)$ �ł���B���̂��Ƃ́A���Εs�ώ� $G(X,Y,Z)$ �� $Z$ �݂̂���Ȃ邱�Ƃ��w���B����A$(GL_1\times SL_4\times GL_2~,~\Lambda_1\otimes\Lambda_1)$ �ɑ��Εs�ώ��͑��݂��Ȃ����Ƃ��m���Ă���B����͖����ł���B�����A$F_2(X,Y,Z)$ �͊���ł���B







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\subsection{nonreg.13}

${\bf (13a)~~(GL_1^2\times SL_7\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda_1\otimes 1~,~Alt_7\oplus Alt_7\oplus V(7))~~N=1}$ \\

�Q�́A$GL_7\times GL_2$ �Ƃ��Ă悢�B$(A,B)\in GL_7\times GL_2$ �ɑ΂��č�p�́A$(X,Y,Z)\longmapsto ((AX{}^tA,AY{}^tA){}^tB,Az)$ �ł���B\\

$1$ �ڂ��l����B \\
$X,Y,Z\in Alt_7$ �ɑ΂��āA\\
\hspace{1cm}$\beta_i(X,Y,Z)=(-1)^{i-1}\{Pf((X+Y+Z)^{(i)})-Pf((X+Y)^{(i)})-Pf((Y+Z)^{(i)}) \\
\hspace{3cm}-Pf((Z+X)^{(i)})+Pf(X^{(i)})+Pf(Y^{(i)})+Pf(Z^{(i)})\}$ \\
�Ƃ����A\\
\hspace{1cm}${}^t\beta(X,Y,Z)=(\beta_1(X,Y,Z)\cdots\beta_7(X,Y,Z))$ �Ƃ����B\\
���̂Ƃ��A\\
\hspace{1cm}$\beta(AX{}^tA,AY{}^tA,AZ{}^tA)=(\det A)~{}^tA^{-1}\beta(X,Y,Z)$ \\
�ł���B\\
$g=(A,B)$ �A $x=(X,Y,z)\in Alt_7\oplus Alt_7\oplus V(7)$ �ɑ΂��āA\\
\hspace{1cm}$F_x(u,v)=\langle\beta(X,X,X)|z\rangle~u^3+3~\langle \beta(X,Y,X)|z\rangle~u^2v \\
\hspace{6cm}+3~\langle \beta(Y,X,Y)|z\rangle~uv^2+\langle \beta(Y,Y,Y)|z\rangle~v^3$ \\
�Ƃ����B
\begin{itemize}
\item $GL_7-action$
\[ \langle\beta(X,X,X)|z\rangle\longmapsto \det A~{}^t\beta(X,X,X)(A^{-1}A)z=\det A~\langle\beta(X,X,X)|z\rangle \]
���A
\[ F_{\rho(A)x}(u,v)=\det A~F_x((u,v)) \]
\item $GL_2-action$ \\
$B=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ �Ƃ����B\\
�܂��A$\Phi(X,Y,z)=\varphi_1u^3+3\varphi_2u^2v+3\varphi_3uv^2+\varphi_4v^3$ �Ƃ����B\\
$\hspace{1cm}\varphi_1 \longmapsto \langle\beta(aX+bY,aX+bY,aX+bY)|z\rangle=a^3\varphi_1+3a^2b\phi_2+3ab^2\varphi_3+b^3\varphi_4 \\
\hspace{1cm}3\varphi_2 \longmapsto 3ac^2\varphi_1+3(bc^2+2acd)\phi_2+3(ad^2+2bcd)\varphi_3+3bd^2\varphi_4 \\
\hspace{1cm}3\varphi_3 \longmapsto 3a^2c\varphi_1+3(bc^2+2abc)\phi_2+3(a^2d+2abd)\varphi_3+b^2d\varphi_4 \\
\hspace{1cm}\varphi_4 \longmapsto c^3\varphi_1+3c^2d\phi_2+3cd^2\varphi_3+d^3\varphi_4$ \\
���A
\[ F_{\rho(B)x}(u,v)=F_x((u,v)B) \]
\end{itemize}
�ȏ���A
\[ F_{\rho(g)x}(u,v)=\det A~F_x((u,v)B) \]
���̂Ƃ��A$(GL_2~,~3\Lambda_1~,~S^3(\mathbb{C}^2))$ �̑��Εs�ώ���p���āA
\[ G(X,Y,z)=P(~\langle\beta(X,X,X)|z\rangle ,3\langle \beta(X,Y,X)|z\rangle,3\langle \beta(Y,X,Y)|z\rangle,\langle \beta(Y,Y,Y)|z\rangle~) \]
�Ƃ����B�A���A\\
\hspace{1.5cm}$P(x_1,x_2,x_3,x_4)=x_2^2x_3^2+18x_1x_2x_3x_4-4x_1x_3^3-4x_2^3x_4-27x_1^2x_4^2$ \\
�ł���B\\
����āA$1$ �ڂ̑��Εs�ώ��́A
\[ F_1(X,Y,z)=G(X,Y,z)\longleftrightarrow (\det A)^4(\det B)^6 \]

�ȏ���A\\
\begin{eqnarray*}
\begin{array}{|c|}\hline
(13a)~~(GL_1^2\times SL_7\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda_1\otimes 1~,~Alt_7\oplus Alt_7\oplus V(7)) \\
\begin{array}{lllll}
F_1(X,Y,z) &=& G(X,Y,z) &\longleftrightarrow& (\det A)^4(\det B)^6
\end{array} \\ \hline
\end{array}
\end{eqnarray*}


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\vspace{1cm}

${\bf (13b)~~(GL_1^2\times SL_7\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda^*_1\otimes 1~,~Alt_7\oplus Alt_7\oplus V(7)^*)~~N=1}$


�Q�́A$GL_7\times GL_2$ �Ƃ��Ă悢�B$(A,B)\in GL_7\times GL_2$ �ɑ΂��č�p�́A$(X,Y,z)\longmapsto ((AX{}^tA,AY{}^tA){}^tB,{}^tA^{-1}z)$ �ł���B\\

$X,Y,Z\in Alt_7$ �ɑ΂��āA\\
\hspace{1cm}$\begin{array}{ccl}\beta_i(X,Y,Z) &=& (-1)^{i-1}\{Pf((X+Y+Z)^{(i)})-Pf((X+Y)^{(i)})-Pf((Y+Z)^{(i)})\\
&&\hspace{1.5cm}-Pf((Z+X)^{(i)})+Pf(X^{(i)})+Pf(Y^{(i)})+Pf(Z^{(i)})\}\end{array}$ \\
 �Ƃ����A\\
\hspace{1cm}${}^t\beta(X,Y,Z)=(\beta_1(X,Y,Z)\cdots\beta_7(X,Y,Z))$ \\
�Ƃ����B���̂Ƃ��A
\[ \beta(AX{}^tA,AY{}^tA,AZ{}^tA)=(\det A)~{}^tA^{-1}\beta(X,Y,Z)\]
�ł���B�����ŁA�ȒP�Ȍv�Z�ɂ��A
\[ \left\{\begin{array}{l}
    \cdot~<\beta(X,X,X)|X|z>=0 \\
    \cdot~3<\beta(X,Y,X)|X|z>~+~<\beta(X,X,X)|Y|z>=0 \\
    \cdot~<\beta(Y,X,Y)|X|z>~+~<\beta(X,Y,X)|Y|z>=0
   \end{array}\right. \]
�����藧�B�����ŁA
\[ \Psi(X,Y,z)=\left(\begin{array}{rr}
-\frac{2}{3} <\beta(X,X,X)|Y|z>&<\beta(Y,X,Y)|X|z> \\
<\beta(Y,X,Y)|X|z>&\frac{2}{3} <\beta(Y,Y,Y)|X|z>
\end{array}\right)\in Sym_2 \]
�Ƃ����B���̂Ƃ��A�ȒP�Ȍv�Z�ɂ��A
\[ \Psi(X,Y,z)\longmapsto (\det A)(\det B)~B\Psi(X,Y,z){}^tB \]
����āA$1$ �ڂ̑��Εs�ώ��́A
\[ F_1(X,Y,z)=\det\Psi(X,Y,z)\longleftrightarrow (\det A)^2(\det B)^4 \]
�ȏ���A\\
\begin{eqnarray*}
\begin{array}{|c|}\hline
(13b)~~(GL_1^2\times SL_7\times SL_2~,~\Lambda_2\otimes \Lambda_1+\Lambda^*_1\otimes 1~,~Alt_7\oplus Alt_7\oplus V(7)^*) \\
\begin{array}{lllll}
F_1(X,Y,z) &=& \det\Psi(X,Y,z) &\longleftrightarrow& (\det A)^2(\det B)^4
\end{array} \\ \hline
\end{array}
\end{eqnarray*}



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�Ō�ɁA�R���s���[�^ $(Mathematica)$ ��p�����\�����@���Љ��B����́A$reg.8$ �y�� $reg.40$ �ɂ����ėp�������@�ł���B�[�I�Ɍ����΁A�i����m�M�̂��Ɓj$generically$\ $surjective$\ $map$ �𖳗��������@�ł���B$reg.8$ ���ɏЉ��B\vspace{.2cm}

${\bf (8)~~(GL_1^2\times SL_5\times SL_8~,~\Lambda_2\otimes \Lambda_1+1\otimes\Lambda^*_1~,~\overbrace{{\bf Alt_5\oplus\cdots\oplus Alt_5}}^8\oplus V(8)^*)~~N=2}$ 

�Q�́A$GL_5\times GL_8$ �Ƃ��Ă悢�B$(A,B)\in GL_5\times GL_8$ �ɑ΂��č�p�́A$(X_1,\cdots,X_8,Y)\longmapsto$\\
$((AX_1{}^tA,\cdots,AX_8{}^tA){}^tB,{}^tB^{-1}Y)$ �ł���B\\

\begin{itemize}
\item $1$ �ڂ��l����B\\
$Y={}^t(y_1,\cdots,y_8)$ �ɑ΂��āA$X\cdot Y:=X_1~y_1+\cdots+X_8~y_8\in Alt_5$ �Ƃ����B\\
���̂Ƃ��A
\begin{eqnarray}\label{reg8-1-1}
X\cdot Y\longmapsto A~(X\cdot Y)~{}^tA
\end{eqnarray}
�ł���B�����ŁA�]���q�s�� $\Delta(X\cdot Y)\in Sym_5$ ���l����B���̂Ƃ��A
\begin{eqnarray}\label{reg8-1-2}
\Delta(X\cdot Y)\longmapsto (\det A)^2~{}^tA^{-1}~\Delta(X\cdot Y)~A^{-1}
\end{eqnarray}
�����藧�B\\

���ɁA$X_i=\left(\begin{array}{ccccc}
0&x^{(i)}_{12}&x^{(i)}_{13}&x^{(i)}_{14}&x^{(i)}_{15}\\
-x^{(i)}_{12}&0&x^{(i)}_{23}&x^{(i)}_{24}&x^{(i)}_{25}\\
-x^{(i)}_{13}&-x^{(i)}_{23}&0&x^{(i)}_{34}&x^{(i)}_{35}\\
-x^{(i)}_{14}&-x^{(i)}_{24}&-x^{(i)}_{34}&0&x^{(i)}_{45}\\
-x^{(i)}_{15}&-x^{(i)}_{25}&-x^{(i)}_{35}&-x^{(i)}_{45}&0
\end{array}\right)\in Alt_5~(i=1,\cdots,8)$ �ɑ΂��āA\\
$\tilde{X_i}=\left(\begin{array}{c}x^{(i)}_{12}\\ \vdots \\ x^{(i)}_{45}\end{array}\right)\in {\bf C}^{10}$ �Ƃ��A$A\in GL_5$ �̍�p�ɑ΂��� $\tilde{X_i}\longmapsto \rho(A)~\tilde{X_i}~(\rho(A)\in GL_{10})$ �Ƃ����B���̂Ƃ��A
$Z=\left[\tilde{X_1},\cdots,\tilde{X_8}\right]\in M(10,8)$ ���l����B\\
�����ŁA$Z$ ����$i$ �s�� $j$ �s $(i<j)$ �������� $8\times 8$ �s��̍s�� $(-1)^{i+j}\det Z^{(ij)}$ �� $ij$ �����Ƃ��� $10\times 10$ ���s��� $\tilde{Z}\in Alt_{10}$ �Ƃ���B\\
���̂Ƃ��A
\begin{eqnarray}\label{reg8-1-3}
\tilde{Z}\longmapsto (\det\rho(A))~(\det B)~{}^t\rho(A)^{-1}~\tilde{Z}~\rho(A)^{-1}
\end{eqnarray}
�ł��邱�Ƃ�������B���� $\det\rho(A)=(\det A)^4$ �ł���B\\
�����ŁA�ȉ��𖞂����悤�ȑΏ̍s�� $\Phi(\tilde{Z})\in Sym_5$ ������B
\begin{eqnarray}\label{reg8-1-4}
\Phi(\tilde{Z})\longmapsto (\det A)^8(\det B)^2~A~\Phi(\tilde{Z})~{}^tA
\end{eqnarray}

�ȏ� $(\ref{reg8-1-2})$ �A$(\ref{reg8-1-4})$ ���A
\begin{eqnarray*}
\Phi(\tilde{Z})~\Delta(X\cdot Y)\longmapsto (\det A)^{10}(\det B)^2A~\Phi(\tilde{Z})~\Delta(X\cdot Y)~A^{-1}
\end{eqnarray*}
�ł���A$1$ �ڂ̑��Εs�ώ��́A
\[��~~~F_1(X_1,\cdots,X_8,Y)=tr\left(\Phi(\tilde{Z})~\Delta(X\cdot Y)\right)\longleftrightarrow (\det A)^{10}(\det B)^2\]

\item $2$ �ڂ��l����B\\

$Y={}^t(y_1,\cdots,y_8)$ �ɑ΂��āA$X\cdot Y:=X_1~y_1+\cdots+X_8~y_8\in Alt_5$ �Ƃ����B\\
���̂Ƃ��A\\
$\hspace{5cm}X\cdot Y\longmapsto A~(X\cdot Y)~{}^tA\hfill(\ref{reg8-1-1})$\\
�ł���B\\
���� $1$ �ړ��l�A$Z=\left[\tilde{X_1},\cdots,\tilde{X_8}\right]\in M(10,8)$ ���l����B\\
�����ŁA$Z$ ����$i$ �s�� $j$ �s $(i<j)$ �������� $8\times 8$ �s��̍s�� $(-1)^{i+j}\det Z^{(ij)}$ �� $ij$ �����Ƃ��� $10\times 10$ ���s��� $\tilde{Z}\in Alt_{10}$ �Ƃ���B\\
���̂Ƃ��A\\
$\hspace{3cm}\tilde{Z}\longmapsto (\det\rho(A))~(\det B)~{}^t\rho(A)^{-1}~\tilde{Z}~\rho(A)^{-1}\hfill(\ref{reg8-1-3})$\\
�ł���A���� $\det\rho(A)=(\det A)^4$ �ł���B\\
�����ŁA�ȉ��𖞂������s�� $\Psi(\tilde{Z})\in Alt_5$ ������B
\begin{eqnarray}\label{reg8-2-1}
\Psi(\tilde{Z})\longmapsto (\det A)^{10}(\det B)^3~{}^tA^{-1}~\Psi(\tilde{Z})~A^{-1}
\end{eqnarray}

�ȏ� $(\ref{reg8-1-1})$ �A$(\ref{reg8-2-1})$ ���A\\
$\hspace{3cm}(X\cdot Y)~\Psi(\tilde{Z})\longmapsto (\det A)^{10}(\det B)^3~A~(X\cdot Y)~\Psi(\tilde{Z})~A^{-1}$\\
�ł���A$2$ �ڂ̑��Εs�ώ��́A
\[��~~~F_2(X_1,\cdots,X_8,Y)=tr\left((X\cdot Y)~\Psi(\tilde{Z})\right)\longleftrightarrow (\det A)^{10}(\det B)^3\]
\end{itemize}

�ȏ���A\\
\begin{eqnarray*}
\begin{array}{|c|}\hline
(8)~~(GL_1^2\times SL_5\times SL_8~,~\Lambda_2\otimes \Lambda_1+1\otimes\Lambda^*_1~,~\overbrace{Alt_5\oplus\cdots\oplus Alt_5}^8\oplus V(8)^*) \\
\begin{array}{lllll}
F_1(X_1,\cdots,X_8,Y) &=& tr\left(\Phi(\tilde{Z})~\Delta(X\cdot Y)\right) &\longleftrightarrow& (\det A)^{10}(\det B)^2 \\
F_1(X_1,\cdots,X_8,Y) &=& tr\left((X\cdot Y)~\Psi(\tilde{Z})\right) &\longleftrightarrow& (\det A)^{10}(\det B)^3
\end{array} \\ \hline
\end{array}
\end{eqnarray*}


�����ŗp�����R���s���[�^ $(Mathematica)$ ��p�����\���̎菇���ȉ��ɋ�����B

�Z$\varphi_{ij}$ �Ɋ܂܂�� $monomial$ ��S�ċ��߂����� $1$ �������ɑ΂��āA$GL_5-action$ �� $equivariant$ �ɂȂ�悤���肷��B���̂Ƃ��A����W���@��p���A�萔�{�������Ĉ�̑������𓾂邱�Ƃ��o����B\vspace{.2cm}\\


\noindent$\tilde{Z}=\left(\begin{array}{cccc}
0&z_{12}&\cdots&z_{110}\\
-z_{12}&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&z_{910}\\
-z_{110}&\cdots&-z_{910}&0
\end{array}\right)\in Alt_{10}$ �ɑ΂��āA�ȉ��̎菇�ɂ��
\begin{eqnarray*}
\Phi(\tilde{Z})\longmapsto (\det A)^8(\det B)^2~A~\Phi(\tilde{Z})~{}^tA
\end{eqnarray*}
�𖞂��� $5\times 5$ �Ώ̍s�� $\Phi(\tilde{Z}):=(\varphi_{ij})_{1\le i,j\le 5}$ ������B\\
���܂��ȉ�����������B
\begin{itemize}
\item $A={\footnotesize\left(\begin{array}{ccccc}a_1&&&&\\&a_2&&&\\&&a_3&&\\&&&a_4&\\&&&&a_5\end{array}\right)}\in GL_5$ �̍�p�őΉ����� $a_1^{s_1}a_2^{s_2}a_3^{s_3}a_4^{s_4}a_5^{s_5}$ �� $z_{ij}$ �̃��X�g���쐬����B����������΁A
\[ \tilde{Z}\longmapsto (\det A)^4~{}^t\rho(A)^{-1}~\tilde{Z}~\rho(A)^{-1} \]
���A�ȉ��̂悤�ȑΉ����X�g�𓾂�B\\
$\hspace{.2cm}\begin{array}{ccc}
z_{12} &\longmapsto& (\det A)^3~a_1^{-1}a_4a_5~z_{12} \\
z_{13} &\longmapsto& (\det A)^3~a_1^{-1}a_3a_5~z_{13} \\
\vdots &\longmapsto& \vdots
\end{array}$
\item $A\in GL_5$ �𕪂��čl����B���Ȃ킿�A$1\leftrightarrow2~,~1\leftrightarrow3,\cdots ,4\leftrightarrow5$ �܂��A$2-s1~,~3-s1,\cdots,5-s4$ ���`�Ɋ�Â��v�Z���A$z_{ij}$ ���e�X�̍�p�Ŕ@���ɕω����邩�̑Ή��\���쐬����B�A���A$1\leftrightarrow2~,~2-s1$ �Ƃ� $A$ �Ƃ��āA${\footnotesize \left(\begin{array}{ccccc}&1&&&\\1&&&&\\&&1&&\\&&&1&\\&&&&1\end{array}\right)}~,~{\footnotesize \left(\begin{array}{ccccc}1&s&&&\\&1&&&\\&&1&&\\&&&1&\\&&&&1\end{array}\right)}$ �Ƃ������̂��w���B
\end{itemize}
�ȏ��p���� $\Phi(\tilde{Z})$ ���v�Z����B\\
��$\varphi_{11}$ �����߂�B
\begin{enumerate}
 \item ��̃��X�g���A $(det A)^{8}a_1^2$ �ɑΉ�����悤�� $2$ ���� $monomial~~z_{ij}~z_{kl}$ ��S�ċ��߁A������ $1$ �������� $T1$ �Ƃ����A$GL_5-action$ �� $equivariant$ �Ȃ��̂��ȉ��ŋ��߂�B
 \item $T1$ �ɑ΂��āA�u�� $1\leftrightarrow2~,~1\leftrightarrow3~,~2\leftrightarrow3~,~4\leftrightarrow5$ �� $equivariant$ �ɂȂ� $1$ ������ $S1$ �����肷��B
 \item $S1$ �ɑ΂��āA $2-\varepsilon1~,~3-\varepsilon1~,~3-\varepsilon2~,~5-\varepsilon4$ �� $equivariant$ �ɂȂ� $1$ �����������肷��B
\end{enumerate}
���u�� $1\leftrightarrow2~,~2-\varepsilon1$ ���ɂ��A���̐����𓾂�B\\
�ȏ�̍�Ƃɂ�� �ȉ��𖞂����悤�ȑΏ̍s�� $\Phi(\tilde{Z})\in Sym_5$ �𓾂�B
\begin{eqnarray*}
\Phi(\tilde{Z})\longmapsto (\det A)^8(\det B)^2~A~\Phi(\tilde{Z})~{}^tA
\end{eqnarray*}

���ʂ��ꕔ������B\\
$\Phi(\tilde{Z})$ �� $1-1$ ���� $\varphi_{11}$ �́A�ȉ��̂悤�ɂȂ�B\\
$\varphi_{11}=z_{510}^2+z_{69}^2+z_{78}^2-2z_{59}z_{610}-2z_{68}z_{79}+2z_{58}z_{710}-2z_{67}z_{89}+2z_{57}z_{810}-2z_{56}z_{910}$\\

�܂��A$\Psi(\tilde{Z})$ �� $1-2$ ���� $\psi_{12}$ �́A�ȉ��̂悤�ɂȂ�B\\
$\psi_{12}=-z_{110}z_{15}z_{210}+z_{110}^2z_{25}-z_{110}z_{19}z_{26}+2z_{17}z_{210}z_{26}+z_{110}z_{18}z_{27}-2z_{16}z_{210}z_{27}-z_{110}z_{17}z_{28}+z_{110}z_{16}z_{29}+z_{15}z_{19}z_{310}-3z_{17}z_{25}z_{310}+2z_{15}z_{27}z_{310}-z_{110}z_{19}z_{35}+z_{17}z_{210}z_{35}+z_{19}^2z_{36}+2z_{27}^2z_{36}-3z_{17}z_{29}z_{36}-z_{18}z_{19}z_{37}-2z_{26}z_{27}z_{37}+z_{17}z_{28}z_{37}+2z_{16}z_{29}z_{37}-2z_{27}z_{35}z_{37}+2z_{25}z_{37}^2+z_{17}z_{19}z_{38}-z_{17}z_{27}z_{38}-z_{16}z_{19}z_{39}+z_{17}z_{26}z_{39}+2z_{17}z_{35}z_{39}-2z_{15}z_{37}z_{39}-z_{15}z_{18}z_{410}+3z_{16}z_{25}z_{410}-2z_{15}z_{26}z_{410}-z_{15}z_{35}z_{410}+z_{110}z_{18}z_{45}-z_{16}z_{210}z_{45}+z_{15}z_{310}z_{45}+2z_{27}z_{36}z_{45}-2z_{35}z_{37}z_{45}-2z_{17}z_{38}z_{45}+2z_{36}z_{45}^2-z_{18}z_{19}z_{46}-2z_{26}z_{27}z_{46}+2z_{17}z_{28}z_{46}+z_{16}z_{29}z_{46}+2z_{25}z_{37}z_{46}-z_{15}z_{39}z_{46}-2z_{26}z_{45}z_{46}-2z_{35}z_{45}z_{46}+2z_{25}z_{46}^2+z_{18}^2z_{47}+2z_{26}^2z_{47}-3z_{16}z_{28}z_{47}+2z_{26}z_{35}z_{47}+2z_{35}^2z_{47}-6z_{25}z_{36}z_{47}+3z_{15}z_{38}z_{47}-z_{17}z_{18}z_{48}+z_{16}z_{27}z_{48}-z_{15}z_{37}z_{48}+2z_{16}z_{45}z_{48}-2z_{15}z_{46}z_{48}+z_{16}z_{18}z_{49}-z_{16}z_{26}z_{49}-2z_{16}z_{35}z_{49}+3z_{15}z_{36}z_{49}+z_{110}z_{12}z_{510}-z_{17}z_{23}z_{510}+z_{16}z_{24}z_{510}-z_{14}z_{26}z_{510}+z_{13}z_{27}z_{510}-2z_{14}z_{35}z_{510}+2z_{13}z_{45}z_{510}-z_{14}z_{210}z_{56}-z_{27}z_{34}z_{56}+z_{24}z_{37}z_{56}+z_{14}z_{39}z_{56}+z_{12}z_{410}z_{56}-z_{34}z_{45}z_{56}+z_{24}z_{46}z_{56}-2z_{23}z_{47}z_{56}-z_{13}z_{49}z_{56}+z_{13}z_{210}z_{57}-z_{12}z_{310}z_{57}+z_{26}z_{34}z_{57}+z_{34}z_{35}z_{57}-2z_{24}z_{36}z_{57}+z_{23}z_{37}z_{57}-z_{14}z_{38}z_{57}+z_{23}z_{46}z_{57}+z_{13}z_{48}z_{57}+z_{110}z_{14}z_{58}+z_{17}z_{34}z_{58}-2z_{14}z_{37}z_{58}-z_{14}z_{46}z_{58}+3z_{13}z_{47}z_{58}-z_{110}z_{13}z_{59}-z_{16}z_{34}z_{59}+3z_{14}z_{36}z_{59}-z_{13}z_{37}z_{59}-2z_{13}z_{46}z_{59}-z_{12}z_{19}z_{610}-z_{15}z_{24}z_{610}+3z_{14}z_{25}z_{610}-z_{12}z_{27}z_{610}-2z_{12}z_{45}z_{610}+z_{24}z_{26}z_{67}-z_{23}z_{27}z_{67}+z_{14}z_{28}z_{67}-z_{13}z_{29}z_{67}-2z_{25}z_{34}z_{67}+z_{24}z_{35}z_{67}+z_{12}z_{39}z_{67}-z_{23}z_{45}z_{67}-z_{12}z_{48}z_{67}-z_{14}z_{19}z_{68}-z_{17}z_{24}z_{68}+2z_{14}z_{27}z_{68}+z_{14}z_{45}z_{68}-3z_{12}z_{47}z_{68}+z_{13}z_{19}z_{69}+z_{17}z_{23}z_{69}-2z_{14}z_{26}z_{69}+z_{15}z_{34}z_{69}-z_{14}z_{35}z_{69}+z_{12}z_{37}z_{69}+2z_{12}z_{46}z_{69}+z_{12}z_{18}z_{710}+z_{15}z_{23}z_{710}-3z_{13}z_{25}z_{710}+z_{12}z_{26}z_{710}+2z_{12}z_{35}z_{710}+z_{14}z_{18}z_{78}+z_{16}z_{24}z_{78}-2z_{13}z_{27}z_{78}-z_{15}z_{34}z_{78}+2z_{12}z_{37}z_{78}-z_{13}z_{45}z_{78}+z_{12}z_{46}z_{78}-z_{13}z_{18}z_{79}-z_{16}z_{23}z_{79}+2z_{13}z_{26}z_{79}+z_{13}z_{35}z_{79}-3z_{12}z_{36}z_{79}-2z_{14}z_{15}z_{810}+2z_{12}z_{17}z_{810}+2z_{14}z_{16}z_{89}-2z_{13}z_{17}z_{89}+2z_{13}z_{15}z_{910}-2z_{12}z_{16}z_{910}$\\
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\begin{thebibliography}{00}
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	\bibitem[K1]{} T.Kimura,\ A\ classification\ of\ prehomogeneous\ vector\ spaces\ of\ simple\ algebraic\ groups\ with\ scalar\ multiplications,{\itshape \/ Journal\ of\ Algebra.}\ {\bf 83},\ No.\ 1~(1983).
	\bibitem[S-K]{} M.Sato\ and\ T.Kimura,\ A\ classification\ of\ irreducible\ prehomogeneous\ vector\ spaces\ and\ their\ relative\ intariants,{\itshape \/ Nagoya\ Math.\ J.}\ {\bf 65}~(1977),\ 1-155.
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    \bibitem[K2]{} �ؑ��B�Y�A�T�ώ��x�N�g����ԁA(1998) �A��g���X
\end{thebibliography}

$Mathematica$ �� $Wolfram\ Research,\ Inc.$ �̓o�^���W�ł��B





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