Complete Positivity of Comultiplication and Primary Criteria for Unitary Categorification

TL;DR

This paper establishes the complete positivity of comultiplication in subfactors and introduces a primary n-criterion for unitary categorification, providing practical tools for analyzing complex multifusion rings.

Contribution

It proves the complete positivity of comultiplication and develops a new primary n-criterion for unitary categorification of multifusion rings, improving practical checkability.

Findings

Primary n-criterion is stronger than Schur product criterion for n≥3.

Criteria can be localized on sparse sets, aiding analysis of high-rank rings.

Numerous examples demonstrate the criteria's effectiveness.

Abstract

In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary $n$-criterion of unitary categorifcation of multifusion rings. It is stronger than the Schur product criterion when $n\geq3$. The primary criterion could be transformed into various criteria which are easier to check in practice even for noncommutative, high-rank, high-multiplicity, multifusion rings. More importantly, the primary criterion could be localized on a sparse set, so that it works for multifusion rings with sparse known data. We give numerous examples to illustrate the efficiency and the power of these criteria.