Addendum to "The Johnson cokernel and the Enomoto-Satoh invariant": The ES-trace detects all top-level partitions

Abstract

The degree $d$ part of the cokernel ${\mathsf{C}}_d$ of the Johnson homomorphism decomposes into irreducible $\mathrm{SP}$-modules indexed by partitions of $d-2r$ for $r\ge 0$: $${\mathsf{C}}_d\cong {\mathsf{C}}_d(d)\oplus {\mathsf{C}}_d(d-2)\oplus \cdots .$$ In this addendum we calculate ${\mathsf{C}}_d(d)$ precisely: it is isomorphic to the $\mathrm{GL}(V)$-decomposition of a space of coinvariants $(V^{\otimes d}{)}_{D_{2d}}$, and the isomorphism is induced by Enomoto and Satoh's trace map. This establishes Conjecture 7.2 of the paper "The Johnson Cokernel and the Enomoto-Satoh invariant."

arXiv commentAddendum to arXiv:1306.3698