Real and quaternionic second-order free cumulants and connections to matrix cumulants

Abstract

We present definitions for real and quaternionic second-order free cumulants, functions whose collective vanshing when applied to elements from different subalgebras is equivalent to the second-order real (resp.\ quaternionic) freeness of those subalgebras. We construct a poset related to the annular noncrossing partitions, and calculate the Möbius function, which may be used to compute the coefficients in the expression for the second-order free cumulants. We show the connection between second-order free cumulants and the topological expansion interpretation of matrix cumulants. This provides a construction for higher-order free cumulants. Coefficients are given in terms of the asymptotics of the cumulants of the Weingarten function.

The third section in previous versions has been moved to paper "Möbius Functions of Some Annular Noncrossing Objects" by the same author, referenced in this paper