New symmetries for Dyson's rank function
Abstract
At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of R(ζp,q), where R(z,q) is the two-variable generating function of Dyson's rank function and ζp is a primitive p-th root of unity. In his lost notebook Ramanujan gives the 5-dissection of R(ζ5,q). This result is related to Dyson's famous rank conjecture which was proved by Atkin and Swinnerton-Dyer. In 2016 the first author showed that there is an analogous result for the p-dissection of R(ζp,q) when p is any prime greater than 3, by extending work of Bringmann and Ono, and Ahlgren and Treneer. It was also shown how the group 1(p) acts on the elements of the p-dissection of R(ζp,q). We extend this to the group 0(p), thus revealing new and surprising symmetries for Dyson's rank function.