Moonshine of Finite Groups by Koichiro Harada

By Koichiro Harada

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11 Theorem. For z 2 H , we have 1 1 D . z/: 1 / z Á. Proof. 1 iz 12 . z/ D e De De iz 12 iz 12 iz 12 i nz 1 X nD 1 1 X nD 1 1 X 2 H , we have /, . 3nC1/ . 12 Exercise. 1 1 z/ : 26 Chapter 2. 13 Exercise. z/ D 1 nD 1. nC1/ (3) D 0 for jxj < 1; nD 1 . z/ D 0. z/ Dp iz 1 X 1 i i 2 12z . 3n n/ Dp e iz nD 1 Dp Dp Dp Hence Á. 1 / z 1 1 1 iz 1 iz 1 iz D . z/. 1 nD1 Á. 1 /: z . 5. a; m/ D 1. mod m/; we say that a is a quadratic residue mod m; otherwise we say that a is a quadratic nonresidue mod m. 14 Definition.

Z/ D C a0 . / C an . /q n ; q nD1 q D e2 iz is ; then the mapping ! an . / from G to C is a generalized character of G. In particular, is a class function of G. G; / for a given group G involves some nontrivial work. € nH / is equal to C. / and that the coefficient an . z/ at 1 are generalized characters of the finite group G for all n 1. N / C for some N . There are exactly 123 possible €’s. Some subgroups, and some conjugates of those 123 €’s are, in practice, the only Fuchsian groups that could be used for a moonshine of a finite group G.

Q/ D iD1 1 X ak . /q k ; kD0 where ak . / is the character of S k . /. q n / D 1 X ak . q / D kD0 ak . /q are characters 0 of G. Now define k . / D ak . / to complete the proof. 29 Exercise. Show P det . / D . 5). Let d be aQdivisor of 24 and be a d dimensional representationQof G over Q. Let D t r t be the frame shape of dh with respect to and D h be a ( fixed ) generalized partition of degree 24=d . z/ D 1 X C ak . /q k ; q kD0 where ak . / are generalized characters of G. , dh > 0 for all h), then all ak .

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