数理解析研究所講究録
第 896 巻 1995 年 53-57

53

REPRESENTATIONS OF MAPPING CLASS GROUPSOBTAINED FROM THE MODULAR INVARIANCE
PROPERTY OF CYCLIC GROUP ASSOCIATION SCHEMESETSUKO BANNAIKYUSHU UNIVERSITY\S 0. Introduction
The concept of the modular invariance property of fusion algebra is very important inconformal field theory. Since Eiichi Bannai found relations between Bose-Mesner algebrasof association schemes and the fusion algebras([l]), many people started to pay attention tothe modular invariance property of association schemes also. It is known that the modularinvariance property of association schemes and generalized spin models have some kind ofrelations([3],[5],[9]) and many examples of generalized spin models are constructed using
the modular invariance property of association schemes([2],[4], etc.). It is also known thatthe spin models give invariants of links (see [10]). In this paper we consider another useof the modular invariance property.
T. Kohno constructed topological invariant of 3-manifold using the modular invarianceproperty of the truncated representations of $sl(2,C)$ (see [11], [12]). He first constructed
projective linear representations of the mapping class groups of Riemann surfaces andthen using Heegaard decompositions of 3-manifolds obtained invariants of 3-manifolds.
Motivated by his work, in this paper we construct some kind of representations of themapping class groups using the modular invariance property of association schemes on finite
cyclic groups. Using the Kohno’s technique we can construct invariants of 3-manifold ffom
those representations. The invariants seem to coincide with the ones obtained by Gocho([8]), however I am not ready to discuss in detail right now so Ijust give the representationsof mapping class groups only(see Theorem 2 in \S 4).

\S 1. Modular invariance property of association schemes
First we introduce the definition of the modular invariance property of association
schemes. Let $X=(X, \{R_{i}\}_{0\leq i\leq d})$ be a self-dual association scheme and let $P$ and be the). For any $i\in\{0,1, \ldots, d\}$first and second eigenmatrices of respectively (note that, whereis the adjacency matrix with respect to the relation
we define by.
$Q$

$X$

$i’$

${}^{t}A_{i}=A_{i}/$$P=\overline{Q}$$A_{i}$$R_{i}$Definition 1 (see [3], [5]) Let be a self-dual association scheme. We say thatmodular invariance property if there exists a diagonal matrix $T=$ diagwhich satisfies one of the following mutually equivalent conditions:
$X$

$X$

has thewith$(t_{0}, t_{1}, \ldots, t_{d})$$t_{0}\neq 0$

Typeset by$A_{J}\backslash \Lambda^{\theta?tX}$

54(1) PTQTPT’ $=t_{0}D^{3}I$ ,(2) PTQTQT $=t_{0}DP^{2}$ ,(3) $(\overline{P}T)^{3}=t_{0}D^{3}I$ ,(4) $(QT)^{3}=t_{0}D^{3}I$ ,(5) $(PU)^{3}=t_{0}^{-1}D^{3}I$ ,where $D^{2}=|X|,$ $T’=$ diag$(t_{0},t_{1’}, \ldots , t_{d’})$and $U=$ diag$(to^{-1}, t_{1}^{-1}, \ldots, t_{d}^{-1})$.

Let $\vee\yen(G_{n})=(G_{n}, \{R_{i}\}_{0\leq i\leq n-1})$ be the association scheme defined on the finite cyclicgroupis self-dual and the character table (firstof order . It is known that
$P=(\zeta^{ij})$with a primitive n-th root of unity. The modulareigen matrix) is given byinvariance property ofis completely determined (see [2], [6]).
Theorem l(see [2])ifwitli a diagonal matrix $T=$ diagand only if the following conditions are satisfied:$G_{n}$$\vee t(G_{n})$$n$$-\yen(G_{n})$$( \frac{1}{\sqrt{n}}PT)^{3}=I$$(\lambda_{0}, \lambda_{1}, \ldots, \lambda_{\tau\iota-1})$,$\lambda_{0^{3}}=\sqrt{n}\eta^{u^{2}}/\sum_{\downarrow=0}^{n-1}\eta^{l^{2}}$for $i\in\{0,1, \ldots , n-1\}$ ,$\lambda_{i}=\eta^{i(i+2u)}\lambda_{0}$where$\eta=\zeta^{\frac{n+1}{2}}$

for

$n$

odd and$\eta^{2}=\zeta$

for

$n$

even and $u\in\{0,1, \ldots, n-1\}$ .

\S 3. Mapping class groupsLet
set of all the orientation
be an orientable surface of gcriiis andbe. The mapping class grouppreserving homeomorphisms ofofis the group. As for the following fact the reader is referred toconsists of all the isotopy cla,sses in$1\leq i\leq g$ ,[7]. [11], [13]. It is kiiown tllat
is generated by the isotopy classes$1\leq i\leq g$ , respectively as $sIiown$ in Fig. 1.
of the Dehn twists about the circles$\Sigma_{9}$$g$$\mathfrak{H}_{9}$$t\}_{1}e$$9J\uparrow_{9}$$\Sigma_{9}$$\Sigma_{9}$$\mathfrak{H}_{9}$$\mathfrak{M}_{9}$$\alpha_{i},$$a_{i},$$b_{i},$$d_{i},$$e_{i},$Fig. 1.
The generating relatioris are given by1,
2 Let and ingeneratirig set.
If the corresponding circles do not intersect, }$ierl$$)$$)$$\delta_{1}=\epsilon_{1}=\alpha_{1_{i}}\delta_{9}=\epsilon_{9}$$\alpha$$\beta$$tI\iota e$$t$$\alpha\beta=\beta\alpha$$\beta_{i},$$\delta_{i},$$\epsilon_{i},$55
If the corresponding circles intersect at one point, }$lerl$$t$$\alpha\beta\alpha=\beta\alpha\beta$

3 If $g=2$ , then$)$,$)$$(\alpha_{1}\beta_{1}\alpha_{2})^{4}=\epsilon_{2}\delta_{2}$5$)$$b_{2}\delta_{2}b_{1}=\alpha_{1}\alpha_{2}\alpha_{3}\delta_{3}$

6$)$

,and

and

$b_{1}=(\beta_{2}\alpha_{2}\alpha_{3}\beta_{2})^{-1}\delta_{2}(\beta_{2}\alpha_{2}\alpha_{3}\beta_{2})$$\delta_{k+1}=J_{k}\delta_{k-1}J_{k}^{-1},2\leq k\leq g-1$

where.

$(\delta_{2}\beta_{2}\alpha_{2}\beta_{1}\alpha_{1^{2}}\beta_{1}\alpha_{2}\beta_{2}\delta_{2})^{2}=1$4where.

and,$b_{2}=(\beta_{1}\alpha_{1}\alpha_{2}\beta_{1})^{-1}b_{1}(\beta_{1}\alpha_{1}\alpha_{2}\beta_{1})$$\epsilon k=G_{k}\delta_{k}G_{k}^{-1},1\leq k\leq g$,$J_{k}=\beta_{k+1}\alpha_{k+1}(\beta_{k}\delta_{k}\alpha_{k}\beta_{k})\alpha_{k+1}\beta_{k+1}\beta_{k-1}\alpha_{k}(\beta_{k}\delta_{k}\alpha_{k+1}\beta_{k})\alpha_{k}\beta_{k-1}$$G_{k}=\beta_{k}\alpha_{k}\cdots\beta_{1}\alpha_{1^{2}}\beta_{1}\cdots\alpha_{k}\beta_{k}$.

\S 4. Representations of the mapping class group
Let be a vector space over the complex nurriber field $C$ wit a basisof andWe define linear isornorphisms $S,$ and$V\otimes V$ in the following manner:
$V$

$f1$

$\Lambda$$T_{u},$$0\leq u\leq\uparrow\iota-1$$S(v_{i})= \frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\zeta^{il}v\iota$$V$$\{v_{0}, u_{1_{\dot{\prime}}}\ldots, v_{n-1}\}$$W_{u},$.

$0\leq u\leq n-1$ , of,$\Lambda(v_{i})=v_{i’}=v_{n-i}$$T_{u}(v_{i})=\lambda_{0}\eta^{i(i+2u)}$,$W_{u}(v_{i}\otimes v_{j})=\lambda_{0}^{-1}\eta^{-(i-j)(i-j+2u)}v_{i}\otimes v_{j}$for $i,j=0,1,$$\ldots,$$n-1$ , where$\zeta$and$(ST_{u})^{3}=I$$\eta$,are given in Theoreni 1. Then we }lavefor any $u\in\{0,\cdot 1, \ldots jn-1\}$ .

Now we define the linear isomorphisms$\overline{\alpha_{i)}}\overline{\beta_{i}},$$\overline{\delta_{i}},$$\overline{\epsilon_{i}},$$1\leq i\leq g$, onby the$V \bigotimes_{9}\cdots\bigotimes_{\tilde{times}}V$

56

following equations:$\overline{\alpha}_{1}=T_{u}^{-1}\otimes I\otimes\cdots\otimes I\sim$$g-1$ times

for $x=2,3,$$\ldots,$$g$,

for $x=1,2,$
for $x=1,2,$
for $x=2,3,$$\overline{\epsilon}_{1}=\overline{\delta}_{1}=\overline{\alpha}_{1}$,$\ldots,$$\ldots,$

$g$

,

$($

$\ldots\prime g$

iiote

,

$tI_{1}at\overline{\delta}_{1}=\overline{\alpha}_{1})$,$g-1_{\dot{J}}$$\overline{\epsilon}_{9}=\overline{\delta}_{9}$$K_{x}(v_{i_{1}}\otimes\cdots\otimes v_{i_{x}}\otimes\cdots\otimes v_{i_{9}})=\zeta^{2ui_{x}}v_{i_{1}}\otimes\cdots\otimes v_{i_{x}}\otimes\cdots\otimes v_{i_{9}}$$P_{x}(v_{i_{1}}\otimes\cdots\otimes v_{i_{x}}\otimes\cdots\otimes v_{i_{9}})=v_{i_{1}}\otimes\cdots\otimes v_{i_{x}-2u}\otimes\cdots\otimes v_{i_{9}}$for $x=1,2,$$\ldots,$$g$,

for $x=1,2_{i}g$ .

We have the following theorem.
Theorem 2 (1) Let and begroups gerierated by$\{\lambda_{0}id,\overline{\Lambda},K_{x},P_{x}|x=1,2,\ldots, g\}$ respectively, where $id$ isand. Then is a normal subgroup of .
$G$$\Gamma$$tI_{1}e$$\{\overline{cx}_{x}, \overline{f}d_{x}, \overline{\delta}_{x}, \overline{\epsilon}_{x}|x=1,\cdot 2, \ldots\dot{}g\}$$t\}_{1}e$identity map of$G$$\Gamma$$\sim V\otimes\cdots\otimes V$9 times(2) If we define a map
,
by
from
to the group,,and homomorphically for other elements in
is a well defined group homomorphism.
$p$$\overline{\beta}_{x}\Gamma$$\rho(\delta_{x})=\overline{\delta}_{x}\Gamma$$G/\Gamma$$\mathfrak{M}_{9}$$\rho(\alpha_{x})=\overline{\alpha}_{x}\Gamma$$\rho(\epsilon_{x})=\overline{\epsilon}_{x}\Gamma$$\rho(\beta_{x})=$. then$\mathfrak{R}t_{g,}$$\rho$

These representations are neither usual representations of groiips nor proje
representations. Actually is a homornorphism from thc mapping class grouporito afactor group of a subgroup in a linear group of finite dimension. It would be iiiteresting if
we could find out the relation with the symmplectic group $Sp(2g, Z)$ .
$(:tive1in\epsilon_{J}^{1}\text{\‘{a}} r$$\mathfrak{M}_{9}$$\rho$REFERENCES[1] E. Bannai, A ssociation schemes and $\int usion$ algeb7 as (an2 (1993), 327-344.
[2] E. Bannai and E. Baiinai, Spi71, $rnodel_{9}$ on finile
torics.

$cy(,\cdot lic$$i7ll7^{\cdot}oducl?O7\iota)$

g7 oups, to, J. of AIgebraic $C^{\backslash }Jornbiriatorics$$\dot{\prime}\iota ppc\backslash \prime aI^{\cdot}i\mathfrak{s}\iota$

J. of$\Lambda lg\epsilon^{Y}1_{3t}\cdot aic\cdot e_{oIItbi\uparrow la-}^{\tau}$

57[3] E. Bannai and E. Bannai, Generalized spin models and association schemes, Memoirs of Fac: Sci.
Kyushu Univ. Ser. A, 47, No. 2 (1993), 397-409.
[4] E. Bannai, E. Bannai, T. Ikuta and K. Kawagoe, Spin models construced from the Harnming association schemes, Proceedings of 10-th Algebraic Combinatorics Symposium at G ifu Univ. (1992),

91-106.
[5] E. Bannai , E. Bannai and F. Jaeger, in preparation.
[6] Etsuko Bannai, Modular invariance property and spm models attached to cyclic group association
schemes, preprint.
[7] J. S. Birman, Mapping class group of surfaces, Contemp. Math. 78 (1988), 13-43.
[8] T. Gocho, The topological invariant of $three-mani \int olds$ based on the $U(l)$ gauge, Proc. JaI)an
Acad. 66A No. 8 (1990), 237-239.
[9] F. Jaeger, Strongly regular graphs and spin models for the Kauffman polynomial, Geom. Dedicata 44(1992), 23-52.
[10] V. F. R. Jones, On knot invariants related to some statislical mechanical models, Pac. i. Math. 137$lh_{CO7}\cdot y$(1989), 311-334.

[11] T. Kohno, Topological invariants for 3-manifolds using representations of mapping class groups I,Topology 31 (1992), 203-230.
[12] G. Moore and Seiberg, Classical and quantam conformal field theory, Comm. Math. Phys. 123 (1989),
177-254.
[13] B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, lsrael J.
Math. 45 (1983), 157-174.

Departmerlt of M athemat ics

Faculty of Science, Kyushu University

Hakozaki 6-10- 1, Higashi-ku

Fukuoka, 812, apan
$J$