When is the Bloch-Okounkov q-bracket modular?

TL;DR

This paper determines when certain quasimodular forms from the Bloch-Okounkov theorem are genuinely modular by analyzing the kernel of a specific operator, providing an explicit basis similar to harmonic polynomials.

Contribution

It introduces a precise condition involving the kernel of an operator to identify when these forms are modular and constructs an explicit basis for this kernel.

Findings

Identifies a condition for quasimodular forms to be modular.

Provides an explicit basis for the kernel of the operator.

Connects the kernel to classical harmonic polynomial spaces.

Abstract

We obtain a condition describing when the quasimodular forms given by the Bloch-Okounkov theorem as $q$-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator {\Delta}. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.