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"Moonshine" of Finite Groups

Koichiro Harada
European Mathematical Society
Publication Date: 
Number of Pages: 
EMS Series of Lectures in Mathematics 12
[Reviewed by
Michael Berg
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This very short book first served as a set of lecture notes for a two month course the author taught at Ohio State about a quarter of a century ago. At that time the “moonshine” of the book’s title was quite novel, very mysterious, and all but irresistible to, e.g., the number theory community. I recall learning about it early on in graduate school (when I was working in automorphic forms), when the mystery in question was phrased in terms of the observation that the Fourier coefficients at the infinite cusp of a certain famous modular form could be expressed as strikingly simple Z-linear combinations of the sizes of the character tables for the irreducible representations of the Monster group. Naturally this had to be a magical phenomenon: not only were modular forms pretty much the hottest thing in number theory, but these were also the days of the flowering, so to speak, of the Gorenstein program for the classification of the finite simple groups. The Griess-Fisher “Monster” was the most prominent among the elusive sporadic simple groups, the last link in the chain. Thus, this flavor of moonshine enjoyed considerable mystique even beyond the borders of finite group theory, algebra, and modular forms and number theory, and what clinched the deal was the association of this movement with none other than the redoubtable John Horton Conway, making for a high profile right out of the starting blocks.

The main moonshine result to be settled at this stage is the Conway-Norton Conjecture, winning Richard Borcherds a Fields Medal. In the Preface to the book under review the author notes that Borcherds’ breakthrough in fact led to the genesis of “a new area of mathematics… the theory of vertex algebras.” Of course, the Conway-Norton Conjecture is precisely concerned with the aforementioned famous modular form (Hilbert’s j-function, because, after all, there is justice in the world) and accordingly qualifies as the spark that set the whole program on its way.

But that’s only the start of the story. While the present author, Koichiro Harada, observes that “the interest generated by researchers in the original mystery of moonshine of the Monster simple group seems to have faded somewhat, due perhaps to the difficulty of solving it,” much of the mystery proper remains and Harada’s objective in launching his book as part of the European Mathematical Society Series of Lectures in Mathematics is to revive interest in this part of number theory and algebra and who knows what else.

So it is, then, that the first two chapters of the book’s four are devoted to, respectively, the needed background material from the theory of modular forms and the Dedekind η-function. But after this, at pretty high speed, Harada gets to moonshine properly so-called and the theme of “[a] multiplicative product of η-functions.” To give an explicit example of what’s going on, here is the critical definition appearing on p. 31: Let F be the collection of all modular functions f of level N (i.e. for a suitable discrete subgroup (Fuchsian, etc.) of SL2(R) containing the Hecke group Γ0(N)) whose compactified fundamental domain is realizable as a genus 0 Riemann surface, whose function field is C(f), and whose Fourier expansion (at the infinite cusp) looks like 1/q + ∑anqn. Then “a pair (G,φ) is ‘moonshine’ for a finite group G if φ is a function from G to F, and if, for σ ∊ G, φσ(z) = 1/q + a0(σ) + ∑an(σ)qn, … then the mapping σ  → an(σ) from G to C is a generalized character of G.” Beautiful isn’t it? That’s the stuff moonshine is made of.

And much of it is still quite mysterious: “Moonshine” of Finite Groups is truly tantalizing and should indeed serve well to seduce fledgling researchers into this field.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

  • Modular functions and modular forms
  • Dedekind eta function
  • "Moonshine" of finite groups
  • Multiplicative product of $\eta$ functions
  • Appendix. Genus zero discrete groups
  • Bibliography