AI 中文总结
该研究引入两类舒尔函数索引的拟模形式族，明确其过渡对应对称函数Hopf代数的卷积，证明特定舒尔艾森斯坦级数构成拟模形式基，并猜想舒尔麦克马洪级数张成带整数系数的所有拟模形式。
AI 中文摘要
我们引入并研究两类由舒尔函数得到的、以分拆为索引的拟模形式族：舒尔艾森斯坦级数与舒尔麦克马洪级数。二者之间的显式过渡可解释为对称函数的Faà di Bruno Hopf代数中的卷积。我们讨论经典的sl₂作用，证明部分大小至多为3的分拆对应的舒尔艾森斯坦级数构成拟模形式的一组基；此外，我们猜想舒尔麦克马洪级数张成所有带整数系数的拟模形式。
英文摘要
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.
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