Symmetric multiple Eisenstein series

Abstract: In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides with the space spanned by modular forms of weight $k$ and the derivative of the Eisenstein series of weight $k-2$. For $k$ odd, we prove that its dimension equals $\lfloor k/3\rfloor$. We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier.