Weingarten integration over noncommutative homogeneous spaces

TL;DR

This paper extends the Weingarten formula to noncommutative homogeneous spaces, providing a framework for probabilistic analysis on these algebraic manifolds under certain symmetry and uniformity assumptions.

Contribution

It introduces a generalized Weingarten formula for noncommutative homogeneous spaces, broadening the scope of integration techniques in noncommutative geometry.

Findings

Established the Weingarten formula for noncommutative homogeneous spaces.

Derived probabilistic consequences from the generalized formula.

Provided axiomatization for the class of spaces considered.

Abstract

We discuss an extension of the Weingarten formula, to the case of noncommutative homogeneous spaces, under suitable "easiness" assumptions. The spaces that we consider are noncommutative algebraic manifolds, generalizing the spaces of type $X=G/H\subset\mathbb C^N$, with $H\subset G\subset U_N$ being subgroups of the unitary group, subject to certain uniformity conditions. We discuss various axiomatization issues, then we establish the Weingarten formula, and we derive some probabilistic consequences.