When is the Bloch–Okounkov q-bracket modular?

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 ∆. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials. * 1 As quasimodular forms were not yet defined, Schoeneberg only showed that θP is modular if P is harmonic. However, for every polynomial P it follows that θP is quasimodular by decomposing P as in Formula (1) below.

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