By Koichiro Harada
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Extra info for Moonshine of Finite Groups
Sample text
2 below. z/ is a primitive cusp form (of some weight, level, and character), then g must satisfy (a)–(d). Proof. Following [8] (and [15]), let us first treat the ‘if’ part of the conjecture. k1 ; k2 ; : : : ; ks / D . 6 of Miyake [18]. 54 Chapter 4. g/ D k1 ks D t1 t` . k1 ; k2 ; : : : ; ks / satisfying the conditions (a)–(d). 24/ D 1575), there are (at most) that many products of Á functions (Á-products). Primitive forms do have the multiplicative property and it is stated in [8] that by examining the first few coefficients (by computer), of the 1575 partitions of 24, all but 30 partitions yield Á-products which are not multiplicative.
In . /; 1Äi1 1 1 "i . /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 .