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

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.