Which compact-group irrep measurements support universal computation?

Fix a compact group $G$, a finite-dimensional unitary representation $V$, and a constant arity $k$. On $N$ systems initially in the maximally mixed state on $V^{\otimes N}$, permit adaptive projective measurements onto the isotypic components of $V^{\otimes r}$ on selected subsets, $r\le k$, with uniform polynomial-time classical control. Preserve coherence within each reported component, including multiplicity. Require an effective description of the representation: each permitted constant-arity instrument must admit a uniform circuit approximation of cost $\operatorname{poly}(\log(1/\varepsilon))$ at error $\varepsilon$, or its implementation is a separately declared oracle resource. Which triples $(G,V,k)$ give universal bounded-error quantum computation under an explicit encoding, which admit efficient classical simulation, and which yield intermediate models? Count all input preparation, readout, ancillary systems, and approximation costs.