Ramifications of Ramanujan's work on \(\eta\)-products
This short note summarizes some observations and calculations relating certain of the sporadic groups with modular forms whose Fourier coefficients are multiplicative (eigenfunctions of Hecke operators). The principal result is based on the fact that the shape of a permutation (the partition of its degree formed by its disjoint cycle lengths) can be used to construct a modular form defined as a product of Dedekind \(\eta\)- functions: for x a permutation, \[ shape(x)=(1^{e_ 1},2^{e_ 2},...,r^{e_ r})\quad \to \quad f_ x(z)=\sum_{t\geq 1}\eta (tz)^{e_ t}. \] It turns out that for all the permutations x in the Mathieu group \(M_{24}\) of degree 24, \(f_ x(z)\) has multiplicative coefficients. Similar observations are made concerning the monster group \(F_ 1\) and further speculation is given concerning the connection of these phenomena with conformal quantum field theory.
scientific article; zbMATH DE number 4134146
scientific article