Transformation and symmetries for the Andrews-Garvan crank function

Abstract

Let $R(z,q)$ be the two-variable generating function of Dyson's rank function. In a recent joint work with Frank Garvan, we investigated the transformation of the elements of the $p$-dissection of $R(\zeta_p,q)$, where $\zeta_p$ is a primitive $p$-th root of unity, under special congruence subgroups of $SL_2(\mathbb{Z})$, leading us to interesting symmetry observations. In this work, we derive analogous transformation results for the two-variable crank generating function $C(z,q)$ in terms of generalized eta products. We consider the action of the group $\Gamma_0(p)$ on the elements of the $p$-dissection of $C(z,q)$, leading us to new symmetries for the crank function. As an application, we give a new proof of the crank theorem predicted by Dyson in 1944 and resolved by Andrews and Garvan in 1988. Furthermore, we present identities expressing the elements of the crank dissection in terms of generalized eta products for primes $p=11,13,17$ and $19$.