Higher Rank TQFT Representations of SL(2,Z) are Reducible

In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum SU(N) for N>2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges SU(N) representation labels by their conjugates. The respective summands of a given parity are typically still reducible into many further components. Specifically, we give an explicit basis for an irreducible direct summand in the SL(2,Z) representation obtained from quantum PSU(3) when the order of the root of unity is a prime r=2 mod 3. We show that this summand is isomorphic to the respective PSU(2) representation.