Procesi's Conjecture on the Formanek-Weingarten Function is False

TL;DR

This paper disproves Procesi's conjecture regarding the monotonicity of the Formanek-Weingarten function, which is related to the generating function for monotone walks on the symmetric group and the Weingarten function of the unitary group.

Contribution

The paper provides a counterexample to Procesi's conjecture, establishing that the conjecture is false and clarifying the properties of the Weingarten function.

Findings

Procesi's conjecture on the monotonicity of the Weingarten function is false.

Counterexamples show the conjecture does not hold in general.

The result clarifies the behavior of the generating function for monotone walks on the symmetric group.

Abstract

In this paper, we disprove a recent monotonicity conjecture of C. Procesi on the generating function for monotone walks on the symmetric group, an object which is equivalent to the Weingarten function of the unitary group.