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 \(R(\zeta _p,q)\) , where R(z, q) is the two-variable generating function of Dyson’s rank function and \(\zeta _p\) is a primitive p-th root of unity. In his lost notebook, Ramanujan gives the 5-dissection of \(R(\zeta _5,q)\) . 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 \(R(\zeta _p,q)\) 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 \(\Gamma _1(p)\) acts on the elements of the p-dissection of \(R(\zeta _p,q)\) . We extend this to the group \(\Gamma _0(p)\) , thus, revealing new and surprising symmetries for Dyson’s rank function.
