Differential Operators and the Wheels Power Series

TL;DR

This paper provides a detailed, self-contained combinatorial proof of the Wheeling isomorphism, connecting algebraic and enumerative combinatorics of graph power series, expanding on previous work related to the Duflo isomorphism.

Contribution

It offers a new combinatorial proof of the Wheeling isomorphism using graph power series and graded averaging maps, differing from earlier algebraic approaches.

Findings

Proof of the combinatorial identity underlying Wheeling.

Demonstration that Wheeling is a graded averaging map.

Extension of algebraic manipulations to combinatorial graph enumeration.

Abstract

An earlier work of the author's showed that it was possible to adapt the Alekseev-Meinrenken Chern-Weil proof of the Duflo isomorphism to obtain a completely combinatorial proof of the Wheeling isomorphism. That work depended on a certain combinatorial identity, which said that a certain composition of elementary combinatorial operations arising from the proof was precisely the Wheeling operation. The identity can be summarized as follows: The Wheeling operation is just a graded averaging map in a space enlarging the space of Jacobi diagrams. The purpose of this paper is to present a detailed and self-contained proof of this identity. The proof broadly follows similar calculations in the Alekseev-Meinrenken theory, though the details here are somewhat different, as the algebraic manipulations in the original are replaced with arguments concerning the enumerative combinatorics of formal power series of graphs with graded legs.