Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in ${\mathcal{M}}_{g,n}$

Abstract

This article investigates the Hodge theory of the moduli space of genus $g$ curves with $n$ marked points, establishing new connections between Schur-Weyl duality for ${\mathfrak{sp}}_g$ and higher Abel-Jacobi invariants. We develop a represe\\ ntation-theoretic framework that decomposes higher Abel-Jacobi invariants of tautological cycles in $C_g^n$ according to symplectic Lie algebra representations, leveraging the Leray filtration and motivic decompositions compatible with ${\mathfrak{sp}}_{2g}$-actions. Central to this work is the introduction of \textbf{higher Faber-Pandharipande cycles} $F{P}_n={\pi }_1^{\times 2}({\Delta }_{12}^n\cdot {\psi }_1)$ in $C{H}^{n+1}(C_g^2)$, a new family of tautological cycles generalizing classical constructions. We prove these cycles are non-torsion under optimal genus constraints: for families over $(n-1)$-dimensional bases, $F{P}_n$ is not rationally equivalent to zero when $g\ge 3n+1$. Furthermore, we determine the precise position of $F{P}_n$ in the Leray filtration of $C_g^2\to M_g$, showing it lies in depth $n+1$ but no deeper, with explicit non-vanishing in $H^{n+1}(M_g,R^{n+1}{f}_{\ast }\mathbb{Q})$ on the $V_{(n+1,1)}$-isotypic component. This yields the first systematic link between Schur-Weyl duality and higher transcendental invariants, revealing that higher diagonals encode geometric phenomena invisible to standard tautological classes.