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 $R(\zeta_p,q)$, 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.Comment: 51 page