Classification of non-homogeneous Fourier matrices associated with modular data up to rank 5

TL;DR

This paper classifies non-homogeneous Fourier and Allen matrices associated with modular data up to rank 5, providing methods that aid in higher rank classifications and insights into recognizing certain $C$-algebras.

Contribution

It offers the first classification of non-homogeneous Fourier and Allen matrices up to rank 5 and introduces techniques for higher rank analysis.

Findings

Classified non-homogeneous Fourier matrices up to rank 5.

Classified non-homogeneous Allen matrices up to rank 5.

Provided criteria to identify $C$-algebras not arising from Allen matrices.

Abstract

Modular data is an important topic of study in rational conformal field theory. A modular datum defines finite dimensional representations of the modular group $\mbox{SL}_2(\mathbf{Z})$. For every Fourier matrix in a modular datum there exists an Allen matrix obtained from the Fourier matrix after dividing each its row with the first entry of that row. In this paper, we classify the non-homogenous Fourier matrices and non-homogenous Allen matrices up to rank $5$. The methods developed here are useful for the classification of the matrices of higher ranks. Also, we establish some results that are helpful in recognizing the $C$-algebras not arising from Allen matrices by just looking at the character table of the $C$-algebra, in particular, the first row of the character table.