AI 中文总结
研究引入多重舒尔级数，通过构建杨表格霍普夫代数研究其代数结构，建立多种关系并关联对称函数环，得到多项式约化公式，还恢复了舒尔多重zeta值等并讨论其（拟）模性。
AI 中文摘要
本文引入多重舒尔级数，它由半标准杨表格上的舒尔型和定义，推广了舒尔多重zeta值和多重艾森斯坦级数。为研究其代数结构，构建杨表格的连通、交换、分次霍普夫代数并将其线性化商与拟洗牌代数等同。在该霍普夫代数及其商中建立了若干关系，还将其与对称函数环关联，得到常数项表格的多项式约化公式。作为应用，恢复了舒尔多重zeta值，引入了舒尔多重艾森斯坦级数及其q模拟，并讨论了它们的（拟）模性。
英文摘要
In this paper, we introduce multiple Schur series, which are defined by Schur-type sums over semi-standard Young tableaux and generalize both Schur multiple zeta values and multiple Eisenstein series. To study their algebraic structure, we construct a connected, commutative, graded Hopf algebra of Young tableaux and identify its linearized quotient with the quasi-shuffle algebra. Within this Hopf algebra and its quotient, we establish several relations, including a hook formula and the Jacobi--Trudi formula. Furthermore, we relate this Hopf algebra to the ring of symmetric functions, which yields polynomial reduction formulas for tableaux with constant entries. As applications, we recover Schur multiple zeta values, introduce Schur multiple Eisenstein series together with a $q$-analogue of Schur multiple zeta values, and discuss their (quasi)modularity.
Comments50 pages
