Schur Eisenstein series and Schur MacMahon series

Technical Summary: Schur Eisenstein Series and Schur MacMahon Series

Problem Statement
The paper addresses the structural relationship between two families of quasimodular forms indexed by partitions: the classical MacMahon sums of divisors (generalized to Schur MacMahon series) and the multiple Eisenstein series (generalized to Schur Eisenstein series). While the relationship between the single-column cases (MacMahon series $A_r$ and all-2 multiple Eisenstein series $G_{\{2\}r}$) was previously established by the first author, a unified framework for arbitrary Young diagrams was lacking. The authors aim to define these generalized families, establish an explicit transition between them, analyze their differential structures under the classical $\mathfrak{sl}_2$-action, and investigate their role in forming bases for the ring of quasimodular forms, particularly regarding integral coefficients.

Methodology
The authors employ the theory of symmetric functions as the unifying language. They define three algebra homomorphisms from the ring of symmetric functions $\Lambda$ (over $\mathbb{Q}$) to the ring of quasimodular forms $\mathcal{M}$:

1. $\Phi_\beta$: Maps power sums $p_r$ to renormalized Schur multiple zeta values $\beta_{2r} = \zeta(2r)/(2\pi i)^{2r}$.
2. $\Phi_g$: Maps $p_r$ to the MacMahon series $P_r = \sum_{m \ge 1} (q^m/(1-q^m)^2)^r$. The image of a Schur function $s_\lambda$ under this map is the Schur MacMahon series $g(\lambda)$.
3. $\Phi_G$: Maps $p_r$ to the renormalized Eisenstein series $G_{2r}$. The image of $s_\lambda$ is the Schur Eisenstein series $G(\lambda)$.

The core methodology involves:

• Hopf Algebraic Transition: Utilizing a deformed Faà di Bruno coproduct $\Delta_F$ on $\Lambda$ to construct an automorphism $\theta$ such that $\Phi_G = \Phi_g \circ \theta$. This interprets the transition between the two families as a convolution in a Hopf algebra.
• Combinatorial Analysis: Using central factorial numbers of the first and second kind to explicitly compute the coefficients of the transition $\theta$.
• Differential Calculus: Defining derivations on $\Lambda$ ($D_\Lambda, W_\Lambda, \delta_\Lambda$) that lift the classical $\mathfrak{sl}_2$-triple action on $\mathcal{M}$ (weight grading, derivative $q \frac{d}{dq}$, and the operator $\delta = -\frac{1}{2}\frac{\partial}{\partial G_2}$).
• 2-adic Divisibility: Employing properties of the Weierstrass differential equation and 2-adic valuations of Eisenstein polynomials to prove linear independence results.

Key Contributions and Results

1. Explicit Transition Theorem (Main Theorem A):
   The authors prove that the Schur Eisenstein series $G(\lambda)$ can be expressed as a linear combination of Schur MacMahon series $g(\mu)$ where $\mu$ is a subdiagram of $\lambda$. Specifically, $G(\lambda) = \beta(\lambda) + \sum_{0 < |\mu| < |\lambda|} d_\lambda(\mu) g(\mu) + g(\lambda)$. The coefficients $d_\lambda(\mu)$ are integers (linear combinations of renormalized zeta values) and are determined by central factorial numbers. This transition is realized via the automorphism $\theta$ of $\Lambda$, which is shown to be the winding automorphism associated with the convolution of the zeta character and the MacMahon specialization.

2. $\mathfrak{sl}_2$-Structure (Main Theorem B):
   The paper constructs a lift of the classical $\mathfrak{sl}_2$-triple $(D, W, \delta)$ on $\mathcal{M}$ to the ring of symmetric functions $\Lambda$.

   • The lowering operator $\delta$ acts on $G(\lambda)$ by summing over all partitions obtained by removing a single box from $\lambda$.
   • The raising operator $D$ acts on $G(\lambda)$ via a determinantal formula involving the column series $G_{\{2\}k}$.
   • The authors show that conjugating this triple by $\theta$ yields a corresponding triple adapted to the Schur MacMahon series $g(\lambda)$, recovering the Andrews–Rose derivative formulas.

3. Basis Theorem (Main Theorem C):
   The authors prove that for any weight $2n$, the set of Schur Eisenstein series $\{G(\lambda) \mid \lambda \vdash n, \lambda_1 \le 3\}$ forms a basis for the space of quasimodular forms $\mathcal{M}_{2n}$.

   • The proof relies on the fact that the transition matrix from the monomial basis $G_2^a G_4^b G_6^c$ to the Schur basis is upper unitriangular modulo 2.
   • This establishes that partitions with parts of size at most 3 are sufficient to generate the entire ring of quasimodular forms, mirroring the fact that $\mathcal{M} = \mathbb{Q}[G_2, G_4, G_6]$.

4. Integral Spanning Conjecture (Main Conjecture D):
   The paper defines the integral lattice $\mathcal{M}_\mathbb{Z} = \mathcal{M} \cap \mathbb{Z}[[q]]$ and the subring generated by Schur MacMahon series with integer coefficients, $\mathcal{G}_\mathbb{Z} = \mathbb{Z}[A_1, A_2, \dots]$.

   • The authors conjecture that $\mathcal{G}_\mathbb{Z} = \mathcal{M}_\mathbb{Z}$, meaning the Schur MacMahon series span the entire integral lattice of quasimodular forms.
   • They provide computational evidence verifying this equality for weights up to 16 and propose a refined bound $B(k)$ for the size of partitions needed to span forms of weight $\le k$.

Significance and Claims
The paper claims to provide a "natural homogeneous correction" of MacMahon's mixed-weight series via Schur Eisenstein series, generalizing previous results from single columns to arbitrary Young diagrams. By interpreting the transition between these families through the lens of a Faà di Bruno Hopf algebra, the authors unify the constant terms (zeta values), the filtered structure (MacMahon series), and the homogeneous structure (Eisenstein series) within a single symmetric function framework.

The significance of the basis theorem lies in identifying a specific, combinatorially natural subset of partitions (those with max part size 3) that generates the entire space of quasimodular forms, offering a new perspective on the algebraic structure of $\mathcal{M}$. The integral conjecture suggests that the Schur MacMahon series are not just a rational basis but a complete integral basis, capturing all divided congruences within the ring of quasimodular forms. The authors note that while the rational generation is proven, the integral equality remains a conjecture supported by computational verification up to weight 16.