Partition Eisenstein series and semi-modular forms

Abstract

1.1. Semi-modular forms. In a landmark 2000 paper [3], Bloch and Okounkov introduced an operator from statistical physics, the q-bracket, under the action of which certain partition-theoretic series transform to quasi-modular forms, a class of functions that includes classical modular forms. This work was expanded on by Zagier [10] and subsequent authors, e.g. [6, 7, 8, 9]. In recent work [4], Bringmann-Ono-Wagner produce families of modular forms via relations to classical Eisenstein series and properties of t-hooks from the theory of integer partitions, again by applying the q-bracket. These works display an intriguing theme: patterns and symmetries within the set P of partitions, give rise to modularity properties. In this paper, we apply similar ideas to answer a theoretical question we pose regarding the existence of classes of special functions: we construct a class of Eisenstein series summed over partitions — and dependent on symmetries within the set P — to produce first examples of what we call “semi-modular forms”, which in short are functions of a complex variable enjoying one of the two canonical invariances of modular forms, as well as a strong complementary invariance property. Let us recall the canonical generators of the general linear group GL2(Z), viz.