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 particular 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.
