Decomposition of some Witten-Reshetikhin-Turaev Representations into Irreducible Factors

TL;DR

This paper analyzes the structure of certain Witten-Reshetikhin-Turaev representations of the mapping class group for genus 2 surfaces, decomposing them into irreducible components for specific levels and exploring some higher genus cases.

Contribution

It provides explicit decompositions of ${ m SU}(2)$ Witten-Reshetikhin-Turaev representations into irreducible factors for particular levels and extends some results to higher genus surfaces.

Findings

Decomposition of representations at levels p=4r, p=2r^2, and p=2r_1r_2.

Identification of irreducible components in these cases.

Partial generalizations to higher genus surfaces.

Abstract

We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are also presented.