Projective representations of Hecke groups from Topological quantum field theory

TL;DR

This paper constructs projective representations of Hecke groups using topological quantum field theory, generalizing modular data and analyzing properties like image group size and reducibility at various levels.

Contribution

It introduces a method to derive projective representations of Hecke groups from TQFT, extending the modular data framework and exploring their algebraic properties.

Findings

The image group of the representation is infinite at low levels in genus 2.

The representation is reducible with at least three irreducible summands at levels 4l+2.

Explicit computations support the properties of the constructed representations.

Abstract

We construct projective (unitary) representations of Hecke groups from the vector spaces associated with the Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces. In particular, we generalize the modular data of Temperley-Lieb-Jones modular categories. We also study some properties of the representation. We show the image group of the representation is infinite at low levels in genus $2$ by explicit computations. We also show the representation is reducible with at least three irreducible summands when the level equals $4l+2$ for $l\geq 1$.