We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar.
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