Another look at the frame shapes of finite groups. (English) Zbl 0653.10024

Harada, Koichiro

J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 491-512 (1987).

The author discusses some transformation formulas, under the action of congruence groups \(\Gamma_ 0(N)\), of certain functions which are products of Dedekind \(\eta\)-functions of the type \(\prod_{t}\eta (tz)^{s_ t}\), where the product is taken over a finite set of positive integers and each \(s_ t\) is an integer. The so-called cyclotomic case has \(t\mid n\) and \(s_ t=\mu (n/t)\) for some integer \(n\). The author proves that the general case can be reduced to the cyclotomic case and gives necessary and sufficient conditions under which these general products are invariant under the group. The action of Atkin–Lehner involutions is also discussed.
(The connection with frame shapes is simply: if \(G\) is the Conway group \(.0\) acting on the Leech lattice, then the characteristic polynomial of the action of an element of \(G\) is a product of cyclotomic polynomials. This determines parameters \(t\) and \(s_ t\) as above and so defines a function of the above type.)

Reviewer: J. L. Hafner

MSC:

11F20	Dedekind eta function, Dedekind sums
11F22	Relationship to Lie algebras and finite simple groups
20D08	Simple groups: sporadic groups

Keywords:

modular functions; transformation formulas; congruence groups; products of Dedekind \(\eta\)-functions; action of Atkin-Lehner involutions; frame shapes; Conway group; Leech lattice; cyclotomic polynomials