数理解析研究所講究録986 巻 1997 年 1-8

1

Geometry of 7 lines on the real projective
plane and the root system of type
$E_{7}$

関口次郎 (Jiro Sekiguchi)

姫路工業大学理学部

The configuration space of 7 points in the real projective plane is
Abstract.
the relationship between the space and the root system ofparticular,In
studied.
is established.
type$E_{7}$

1. Introduction and the main results
The main purpose of this article is to clarify the relationship between the geometry of 7.
and the root system of typelines on the real projective plane. We give conditions on theseon
We first introduce marked 7 lines
7 lines:$E_{7}$$\mathrm{P}^{2}(\mathrm{R})$$(l_{1}, l_{2}, \cdots, l_{7})$$\mathrm{P}^{2}(\mathrm{R})$, are mutually different.
1. The 7 lines
intersect at a point.
2. No three of.
3. There is no conic tangent to six of$l_{1},$$l_{2},$$l_{1},$$l_{2},$$\cdots$$\cdots,$$l_{7}$$l_{7}$$l_{1},$$l_{2},$$\cdots,$$l_{7}$

with conditions 1, 2 forms the configuration
The totality of marked 7 lines on
space $P(3,7)$ : the space $P(3,7)$ is defined by$\mathrm{P}^{2}(\mathrm{R})$$P(3,7)=GL(3, \mathrm{R})\backslash M/(3,7)/(\mathrm{R}^{\cross})^{7}$,

real matrices of which no 3-minor vanishes. On the otherwhere $M’(3,7)$ is the set ofwith conditions 1, 2, 3 forms a subset ofhand, the totality of marked 7 lines on.
$P(3,7)$ which we denote by $P_{0}(3,7)$ . Both $P(3,7),$ $P_{0}(3,7)$ are affine open subsets of
induce a biregular -action on $P(3,7)$ (and also
Permutations on the 7 lines
on $P_{0}(3,7)$ is naturally extended
that on $P_{0}(3,7))$ . It is stressed here that thebe the set of connected components of
to a biregular $W(E_{7})$ -action (cf. [3], [4]). Let. Then we are in a
$P_{0}(3,7)$ .
The $W(E_{7})$ -action on $P_{0}(3,7)$ naturally induces that on
position to state one of the main results of this article.
$3\cross 7$$\mathrm{P}^{2}(\mathrm{R})$$\mathrm{R}^{6}$$l_{1},$$l_{2},$$\cdots,$$S_{7}$$l_{7}$$S_{7^{-\mathrm{a}}}.\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$P_{7}$$P_{7}$

Theorem 1 The $W(E_{7})$ -action on P7 is transitive.
Before entering into the main text, we are going to explain the motivations of this study
briefly. It is an interesting problem to construct a tame compactification of the configura.
of marked hyperplanes on the complex projective spacetions space$P\mathrm{c}^{(n,k)}$$k$$\mathrm{P}^{n-1}(\mathrm{C})$2, the so called Terada
In the case of $n=2$ , there is a nice compactification ofadmits -action. What happens if we treat $P(2, k)$model. We note here that
. In [8], it, where $P(2, k)$ is the configuration space of -points of
instead ofis shown that the -action on the totality of connected components of $P(2, k)$ is transitive
and that each connected component in question is described in terms of “juzu” introduced
there. On the other hand, in the case $(n, k)=(3,6)$ , there is a tame compactification of thewith conditions 1, 2, 3 above constructedconfiguration space of marked six lines ofin [2] which is called Naruki’s cross ratio variety and denoted by . As an application of
the results of [2], [3], already shown is the result which we are going to explain (cf. [7]).
Let $P_{0}(3,6)$ be the space defined similarly to $P_{0}(3,7)$ , changing the 7 lines with 6 lines.
We note that $P_{0}(3,6)$ admits a $W(E_{6})$ -action (cf. [2], [3]). Then the result for the case
$P_{0}(3,6)$ given in [8] similar to the case of Terada models is that there are 432 connected
itive. Theorem 1 iscomponents in $P_{0}(3,6)$ and that the $W(E_{6})$ -action on the set isan analogue of these results to the case of 7 lines on the real projective plane.
The second main result is the classification of the -orbital structure of . This will bedone in terms of tetradiagrams for the root system of type . In particular, we conclude
that P7 is decomposed into 14 -orbits (cf. Theorem 4).
$P\mathrm{c}(2, k)$$P\mathrm{c}(2, k)$$S_{k}$$k$$P\mathrm{c}(2, k)$$\mathrm{P}^{1}(\mathrm{R})$$S_{k}$$\mathrm{P}^{2}(\mathrm{C})$$C$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\dot{\mathrm{s}}$$\ddot{P}_{7}$$S_{7}$$E_{7}$$S_{7}$

2. Review on the root system$\triangle$of type$E_{7}$

We first recall the definition of the root system of type
. Let be an 8-dimensional. Let
be the inner product on
Euclidean space with a standard basis
defined by$\tilde{E}$$E_{7}$$\{\epsilon_{j;}1\leq j\leq 8\}$$\langle\cdot, \cdot\rangle$$\tilde{E}$$\langle\epsilon_{j}, \epsilon_{k}\rangle=\delta_{jk}$and let $E$ be its linear subspace orthogonal tovectors of $E$ :$\epsilon_{7}+\epsilon_{8}$. We define the following sixty-three,$\gamma_{1}$$=$$\epsilon_{8}-\epsilon_{7}$$\gamma_{j}$$=$$-\epsilon_{j-1}+\gamma_{0}$$\gamma_{1j}$$=$$-\epsilon_{j-1}+\gamma_{0}$$\gamma_{jk}$$=$$\mathcal{E}_{j-}-1^{-\epsilon}k1$$\gamma_{1jk}$$=$$-\epsilon_{j-1^{-}}\epsilon\iota.-1$$\gamma_{ijk}$$=$$-\epsilon_{i-1}-\epsilon j-1-\epsilon_{k}-1+\gamma_{0}$,,,$1<j<8$$1<j<8$$1<j<k<8$$1<j<k<8$,$1<i<j<k<8$

,

where
$\gamma_{0=\frac{1}{2}}\sum_{j=1}\epsilon j-\mathcal{E}_{7}8$.

Note that$\gamma_{i}\perp\gamma_{jk}$$\dot{\mathrm{T}}$he totality,$\gamma_{i}\perp\gamma_{ijk}$$\Delta \mathrm{o}\mathrm{f}\pm\gamma_{j},$$\pm\gamma_{jk},$,.

$\gamma_{ij}\perp$$\pm\gamma_{ijk}$$\gamma_{12}$,$\gamma_{kl}$,$\gamma_{ij}\perp\gamma_{ijk}$,

forms a root system$\gamma_{123}$,$\gamma_{23}$,$\gamma_{34}$,$\gamma_{45}$,$\gamma_{ij}\perp\gamma klm$

$0.\mathrm{f}$$\gamma_{56}$

type

,

$\gamma_{67}$$E_{7}$’$\gamma_{ijk}\perp\gamma_{ilm}$. It is clear that(1)

3
can serve as a system of positive simple roots; its extended Dynkin diagram is given as$–$$\gamma_{1}$$\gamma_{12}$

$–$

$–$

$\gamma_{23}$$–$$\gamma_{34}$$–$$\gamma_{45}$$\gamma_{56}$$–$$\gamma_{67}$

(2)

$|$

$\gamma_{123}$

The setLet
on as

is thepositive roots of .
be the reflections onwith respect to$\triangle$$\{\gamma_{i}, \gamma_{j\gamma_{i}jk}k,\}$$s_{i},$$S_{ij},$$S_{ij}k$$\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}_{0}.\mathrm{f}$$E$$\gamma_{i},$$\gamma_{ijk}$$\gamma_{ij},$. These reflections act$\Delta$$s_{i}$:$\gamma_{j}rightarrow\gamma_{ij}$,$\gamma_{jk}rightarrow\gamma_{jk}$$s_{ij}$$s_{123}:’\gamma 1rightarrow\gamma_{1},$,$\gamma_{jkl}rightarrow\gamma_{mnp}$,$\{i,j, k, l, m, n,p\}--\{1,2, \ldots, 7\}$

: permutation of the indices$\gamma_{4}rightarrow\gamma_{567},$$\gamma_{12}rightarrow\gamma_{12},$$i$$\gamma_{14}rightarrow\gamma_{234},$

,

and ,
$j$

$\gamma_{45}rightarrow\gamma_{45},$.

$\gamma_{145}rightarrow\gamma_{145}$

modulo signs. We define two reflection groups$G_{1}=\langle s12, S_{23}, s_{34}, s_{45}, S_{5}6, s67\rangle\cong S_{7}$$G=\langle G1, s123\rangle\cong W(E_{7})$,,is the symmetric group on seven numerals $\{1, 2, \ldots, 7\},$ $W(E_{7})$ the Weyl groupwhere
denotes the group generated by
Note that
andacts onof typetransitively on .
$S_{7}$$E_{7}$$a,$ $b,$$\langle a, b, \ldots\rangle$$\ldots$$G$$\Delta$$\Delta$

3. Connected components of
We start with introducing a$3\cross 7$$C_{\mathrm{R}}’(\triangle, D_{4})$matrix$X=$.

Then by taking the entries of the vector $(1 x y)\cdot X$ , we obtain seven lines of the real
as inhomogeneous coordinates:
projective plane regarding$x,$ $y$$L_{1}$: $x+y+t=0$$L_{2}$:. $x=0$$L_{3}$:$L_{4}$$L_{5}$$L_{6}$$L_{7}$

We assume that

$t$

$y=0$

: $x+y+$. $1=0$: $-2x+4y+1=0$: $5x-y+1=0$: $6x-6y+1=0$

is real and sufficiently large.

(3)

4
By the seven lines above, we obtain ten triangles $(T_{j})(j=1,2, \cdots, 10)$ surrounded by
the three lines(4)

Corresponding to these ten triangles, there is a 6-polytope
going to define.

$P$

of

$C_{\mathrm{R}}’(\triangle, D_{4})$Theorem 2 There is a simply connected and connected component $P$ ofrounded by the following hypersurfaces:

which we are$C_{\mathrm{R}}’(\triangle, D_{4})$

sur-

(5)

be the totality of these hypersurfaces. Then, for each $w\in W(E_{7}),$ $w\cdot P$ isLet.
surrounded by the hypersurfaces contained inbe the reflections with respectof $P$ in $W(E_{7})$ . Let
We consider, respectively. Using this notation, we put
to the roots$\mathcal{H}(P)$$w\cdot \mathcal{H}(P)$$\mathrm{a}\mathrm{u}\mathrm{t}_{\mathrm{o}\mathrm{n}}1\mathrm{o}\mathrm{r}_{\mathrm{P}}\mathrm{h}\mathrm{i}_{\mathrm{S}}\mathrm{m}\mathrm{s}$$\gamma_{i},$$\gamma_{i}j,$$S_{i},$$S_{ij},$$S_{ij}k$$\gamma ijk$$\sigma_{1}=s_{12}s_{34}s57S_{6}$$\sigma_{2}=s_{35^{S_{24^{S}562}}}s$,.

Then it is easy to show that$\sigma_{1}\cdot P=\sigma_{2}\cdot P=P$.

is isomorphic to. Thenbe the group generated bygroup on four letters. This is proved as follows. First we note thatLet

$G_{P}$$\sigma_{1},$$\sigma_{1}^{2}=1$,$G_{P}$$\sigma_{2}$$\sigma_{2}^{3}=1$,$(\sigma_{1}\sigma_{2})^{4}=1$.

Putting$-1$$\tau_{1}=\sigma_{2}\sigma_{1}\sigma_{2}$,$\tau_{2}=\sigma_{1}$,$-1$$\tau_{3}=\sigma_{1}\sigma_{2}$$\sigma_{1}\sigma_{2}\sigma_{1}$,$S_{4}$, the symmetric5
we find$\mathrm{t}\mathrm{h}\mathrm{a}\grave{\mathrm{t}}$the correspondence$\tau_{1}arrow(12)$induces the isomorphism between, it follows that
$G_{P}=<\mathcal{T}_{1},$$\sigma_{2}^{-1}$Lemma 1 By the$G_{P}$$\mathcal{T}_{2},$,$\tau_{2}arrow(23)$$<\tau_{1},$$\mathcal{T}3>$,$\tau_{3}arrow(34)$$\tau_{2,3}\mathcal{T}>\mathrm{a}\mathrm{n}\mathrm{d}S_{4}$. On the other hand, since$\tau 2\tau 1\tau 2\tau 3=$.

-action, the $hyperSurfaCe\mathit{8}$ in$A_{j}$

are $tran\mathit{8}itive(j=1,2,3,4,5,6)$ .

acts on the 6-polytope
On the other hand, the center of $W(E_{7})$ which is isomorphic to$P$ as the identity transformation. Noting thatconsists of four hypersurfaces, we obtainthe following.
$\mathrm{Z}_{2}$$A_{2}$

Proposition 1 The isotropy subgroup of$P$

in $W(E_{7})$ is isomorphic to the group$S_{4}\cross \mathrm{Z}_{2}$.

4. 6-polytopes adjacent to $P$
To study the 6-polytopes adjacent to $P$ , it is better to treat $P’=s_{34^{S}23}s_{23}6S16s_{5}7^{\cdot}P$
instead of $P$ . Then $P’$ is surrounded by the hypersurfaces in TABLE I below:
TABLE I

(6)

Lemma 2 The foltowing hold.
$\mathcal{H}(P’)\cap \mathcal{H}(s12^{\cdot}P’)$$\mathcal{H}(P’)\cap \mathcal{H}(s_{6}7^{\cdot}P’)$$\mathcal{H}(P’)\cap \mathcal{H}(s13^{\cdot}P’)$$=$$\{H\in \mathcal{H}(P’);s12^{\cdot}H=H\}$$=$$\{H\in \mathcal{H}(P’);\overline{H}\cap\overline{Y_{1}2}\neq\emptyset\}$$=$$\{H\in \mathcal{H}(P’);S67^{\cdot}H=H\}$$=$$\{H\in \mathcal{H}(P’);\overline{H}\cap\overline{Y67}\#\emptyset\}$$=$$\{H\in \mathcal{H}(P’);S13^{\cdot}H=H\}$$=$$\mathcal{H}(P’)\cap \mathcal{H}(s14S_{2}3^{\cdot}P’)$$\mathcal{H}(P’)\cap \mathcal{H}(s_{4}7s_{56}\cdot P’)$$\mathcal{H}(P’)\cap \mathcal{H}(us_{1}\cdot P’)$$\{H^{\cdot}\in \mathcal{H}(P’);\overline{H}\cap\cdot\overline{z_{1}23}\neq\emptyset\}$$=$$\{H\in \mathcal{H}(P’);s14^{S_{2}}3^{\cdot}H=H\}$$=$$\{H\in \mathcal{H}(P’);\overline{H}\cap\overline{W567}\neq\emptyset\}$$=$$\{H\in \mathcal{H}(P’);s_{4}7s_{5}6^{\cdot}H=H\}$$=$$\{H\in \mathcal{H}(P^{J});\overline{H}\cap\overline{W123}\neq\emptyset\}$$=$$\{H\in \mathcal{H}(P’);uS_{1}\cdot H=H\}$$=$$\{H\in \mathcal{H}(P’);\overline{H}\mathrm{n}\overline{x_{0}}\neq\emptyset\}$where $u=(1765432)\in W(E_{7})$ .

6As a consequence of Lemma 2, we find the following.

Theorem 3 The following statements hold:(i) The 6-polytopes $P’$ andare adjacent to each other and(ii) The 6-polytopes $P’$ andare adjacent to each other and$P’$(iii) The 6-polytopes
andare adjacent to each other and. (iv) The 6-polytopes $P’$ andadjacent to each other$P’$(v) The 6-p..olytopes
adjacent to each otherand(vi) The
and $us_{1}\cdot P’a.re$ adjacent to each other and$\overline{P’}\cap s_{12}\cdot\overline{P’}\subset Y_{12}$$s_{12}\cdot P’$.

$\overline{P’}\cap s_{67}\cdot\overline{P’}\subset Y_{67}$$s_{67}\cdot P’$$s_{13}\cdot P’$$s_{14}s_{23}\cdot P/are$.

$\overline{P’}\cap s_{13}\cdot\overline{P’}\subset Z_{123}$.

$and \overline{P’}\bigcap_{S}14s23^{\cdot}\overline{P’}\subset W_{567}$$s_{47}s_{56}\cdot P/are$$and\overline{P’}\cap S_{47^{S\cdot\overline{P}}}56’\subset W_{123}$$\mathit{6}- p_{\mathit{0}}lyt_{\mathit{0}}pe\mathit{8}P/$$\overline{P’}\cap uS_{1}\cdot\overline{P’}\subset X_{0}$.
.

.

Theorem 1 is a consequence of the theorem above.

5. Diagrams corresponding to 6-polytopes
Let $P’$ be the 6-polytope introduced in the previous section. From TABLE I, we findthat there are 10 hypersurfaces incorresponding to positive roots of typebelow:$A_{1},$$A_{2}$$E_{7}$TABLE II
(7)

From these 10 roots, it is possible to construct a diagram similar to Dynkin diagrams.
Namely, the diagram in question consists of 10 circles attached with the 10 roots in TABLE
II. Two circles are connected by a segment if and only if the roots attached to them are
not orthogonal each other. Then we find that there are 12 segments in this diagram.
In the same way, to each 6-polytope in, there associates a diagram which is called atetradiagram for the reason that in the case of $P’$ , the 4 circles corresponding to the roots inof TABLE II are regarded as 4 vertices of a tetrahedron and the 6 circles corresponding
to the roots inof TABLE II are regarded as 6 middle points of the sides of the referred
tetrahedron.
To avoid the complexity, we neglect theof tetradiagrams. In this manner, wefinally obtain a tetra-diagram corresponding to each
. We denote by
the setof tetra-diagrams.
We here note that the classification of the arrangements of 7lines
on the realprojective plane with conditions 1, 2, 3 is accomplished if we determine the decompositionof
into -orbits. This will be done in the next section.
$P_{7}$$A_{2}$$A_{1}$$\pm \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{s}$$P^{\prime J}\in P_{7}$$\mathcal{T}P_{7}$$(l_{1}, l_{2}, \cdots, l_{7})$$\mathcal{T}P_{7}$$S_{7}$

6. Tetradiagrams
We have already introd\"uced tetradiagrams in the previous section. In this section, we
treat them in the language of root systems systematic manner.

7
Definition 1 Let $a_{i}(i=1,2,3,4),$ $b_{ij}(.1\leq i<j\leq 4)$ be roots..of
for all $i,j.$ ) Then the- set$\Delta$. (Assume that$b_{ij}=b_{ji}$$\mathrm{A}=\{a_{i};i=1,2,3,4\}\cup\{b_{ij;}1\leq i<j<\leq 4\}$is called a tetrahedral set if the following conditions hold:$(i\neq j)$ .
(i)$(\{i,j\}\neq\{i’,j’\})$ .
(ii)
.
(iii)
if and only if$\langle a_{i}, a_{j}\rangle=0$$\langle b_{ij,j’}b_{i}t\rangle=0$$i\not\in\{j, k\}$$|\langle a_{i}, b_{jk}\rangle|=0$

Example 1 The set$\cup=\{\gamma_{3}45, \gamma 123, \gamma_{136}, \gamma 256, \gamma_{1}35, \gamma 167, \gamma 347, \gamma 124,(.A\mathrm{c})\dot{\mathrm{O}}, \gamma_{2}\wedge 75\}$

is a tetrahedral set. In particular, the correspondence$\gamma_{345}$$arrow$$\mathrm{A}_{1}$$\gamma_{123}$$arrow$$\mathrm{A}_{2}$$\gamma_{146}$$arrow$

A3$\gamma_{256}$$arrow$$\mathrm{A}_{4}$$\gamma_{135}$$arrow$$\mathrm{B}_{12}$$\gamma_{167}$$arrow$$\mathrm{B}_{13}$$\gamma_{347}$$arrow$$\mathrm{B}_{14}$$\gamma_{124}$$arrow$$\mathrm{B}_{23}$$\gamma_{236}$$arrow$$\mathrm{B}_{24}$$\gamma_{257}$$arrow$

B34

induces a tetradiagram for U.
For a tetrahedral set$\mathrm{A}=\{a_{i}\}\cup\{b_{ij}\}$, we put$\tilde{\mathrm{A}}=\{\pm a_{i}\}\cup\{\pm bij\}$

be also a tetrahedral set. Then A andand call it an extended tetrahedral set. Let. In this case, we confuse a tetradiagram for A and thatare equivalent if and only if
for , for simplicity.
-orbit structure will be also important. For this purpose, we will introduceThe
fourteen extended tetrahedral sets in TABLE III. If A is a tetrahedral set whose extended
the -orbit of . If is atetrahedral set has the name X in TABLE III, we denote by, we call a tetrahedral set of type X. Similarly, we calltetrahedral set such thata tetradiagram for that of type X.
tetrahedral sets.
We are going to give a classification of -orbital structure of
For this purpose, we define$\mathrm{A}’$$\mathrm{A}’$$\overline{\mathrm{A}}=\tilde{\mathrm{A}}’$$\mathrm{A}’$$G_{1}$$\tilde{\mathrm{A}}$$\tilde{\mathrm{B}}\in O_{\mathrm{X}}$$\mathrm{B}$$S_{7}$$O_{\mathrm{X}}$$\mathrm{B}$$\mathrm{B}$$\dot{\mathrm{e}}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}- \mathrm{e}\mathrm{d}$$S_{7}$$L=$

Theorem 4 The set.

$\mathcal{T}${ , Bl, ..., B5, Cl, ..., C4, DI, ...D4}.
$\mathrm{A}$of extended tetrahedral sets is decomposed into fourteen$S_{7}$-orbits$O_{\mathrm{X}}(X\in L)$

are isomorphic. Therefore the classification
It is clear from the definition that andof -orbits of is accomplished by Theorem 4.
$\mathcal{T}$$S_{7}$$\mathcal{T}$$T\mathcal{P}_{7}$8TABLE III” ’,}$\ovalbox{\tt\small REJECT}_{\mathrm{c}}^{\mathrm{A}}\mathrm{c}_{4}\mathrm{D}\pm\{\gamma 7,\gamma 14’ 6,’\gamma’,’\gamma’ 1’ 3’,’,\gamma’,’,’\gamma 1" 4,"\gamma_{1}",,’\gamma_{2}"" 4\}’$

$,$

(Here

$\pm\{\gamma_{345},$$\gamma_{1}23,$$\gamma_{1}46,$$\ldots\}$

is an abbreviation of$\{\pm\gamma_{345},$ $\pm\gamma_{123},$ $\pm\gamma_{146},$$\ldots\}.$)

参考文献[1] N. Bourbaki: Groupes et alg\‘ebre de Lie, Chap. IV- VI. Hermann, Paris, 1968.
[2] I. Naruki: Cross ratio variety as a moduli space of cubic surfaces. Proc. London Math. Soc.
45 pp. 1-30 (1982).
[3] J. Sekiguchi: Cross ratio varieties for root systems. Kyushu J. Math. 48 pp. 123-168 (1994).
[4] J. Sekiguchi: Cross ratio varieties for root systems. II. Preprint.
[5] J. Sekiguchi: Cross ratio varieties for root systems of typeSci. Univ. Tokyo 3 pp. 181-197 (1996).

$A$

and the Terada model. J. Math.

[6] J. Sekiguchi and T. Tanabata: Tetradiagrams for the root system of type
cation. Preprint.

$E_{7}$and its appli-[7] J. Sekiguchi and M. Yoshida: The $W(E_{6})$ -action on the configuration space of six points of
the real projective plane. Preprint.

[8] M. Yoshida: Democratic compactification of the configuration spaces of point sets on thereal projective line. Kyushu J. Math. 50 pp. 493-512 (1996).