Schur Eisenstein series and Schur MacMahon series

Abstract

We introduce and study two partition-indexed families of quasimodular forms obtained from Schur functions: Schur Eisenstein series and Schur MacMahon series. An explicit transition between them can be interpreted as a convolution in a Faà di Bruno Hopf algebra of symmetric functions. We discuss the classical sl2-action and prove that Schur Eisenstein series for partitions with parts of size at most 3 give a basis for quasimodular forms. Further, we conjecture that the Schur MacMahon series span all quasimodular forms with integral coefficients.

Disclosure

“ree with (4.16), and the obstruction to the naive bound B(k) = k/2 is 3-primary in every computed case. AI & computational resource disclosure: The main results, their formulation, and the underlying idea of proof are the authors’ own. ChatGPT 5.6 and Claude Fable 5 were used throughout as research assistants: they carried out and checked computations, and helped with implementing all the objects in SageMath. All statements and proofs were verified by the authors, who take sole”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T19:13:04.043176Z

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