Traces of partition Eisenstein series

Abstract

We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(\ddot{I}„),$ defined by $$\hat{I}»=(1^{m_1},2^{m_2},\dots ,k^{m_k})⊢k⟼G_{\hat{I}}»(\ddot{I}„):=G_2(\ddot{I}„{)}^{m_1}{G}_4(\ddot{I}„{)}^{m_2}\cdots G_{2k}(\ddot{I}„{)}^{m_k}.$$ For functions $\ddot{I}•:\mathcal{P}\to \mathbb{C}$ on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form $${\mathrm{Tr}}_k(\ddot{I}•;\ddot{I}„):=\sum _{\hat{I}»⊢k}\ddot{I}•(\hat{I}»)G_{\hat{I}}»(\ddot{I}„).$$ These traces give explicit formulas for some well-known generating functions, such as the $k$th elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}\ddot{I}„,$ as well as the $2k$th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.

A few typos fixed