The Bloch-Okounkov theorem yields a $\eufm{sl}_2$-equivariant map defined by sums over partitions from shifted symmetric polynomials to quasimodular forms. Classical modular forms are precisely given by the kernel of one of the operators of this $\eufm{sl}_2$-triple on quasimodular forms. The inverse image of modular forms, which is the kernel of another operator $\Delta$, is called the space of harmonic shifted symmetric polynomials. We find an explicit basis for this space using an analogue of the Kelvin transform from the theory of harmonic functions.
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[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/7866