Algebraic proof of the ascension-function recursion

Prove algebraically the convolution identity relating the unitary Weingarten functions in consecutive dimensions, namely \(\Wg_{k,n}=\Raise_{k,n}*\Wg_{k,n-1}\) for \(k+1\le n\), without using random-matrix arguments.

Background

Theorem 3.1 establishes the recursion $\Wg_{k,n}=\Raise_{k,n}*\Wg_{k,n-1}$ by decomposing a Haar-distributed unitary matrix into a complex reflection and an independent lower-dimensional Haar unitary matrix. The proof is probabilistic and relies on the virtual-isometry construction.

Because the resulting identity is an algebraic convolution equation in the symmetric-group algebra, the authors explicitly point out that a direct algebraic proof remains to be found. Such a proof could illuminate the structural role of the ascension function independently of the random-matrix realization.