Decompositions of small tensor powers and Larsen's conjecture

We classify all pairs $(G,V)$ with $G$ a closed subgroup in a classical group $\mathcal G$ with natural module $V$ over $\mathbb C$, such that $\mathcal G$ and $G$have the same composition factors on $V^{\otimes k}$ for a fixed $k\in \{2,3,4\}$. In particular, we prove Larsen's conjecture stating that for $\dim(V)>6$ and $k=4$ there are no such $G$ aside from those containing the derived subgroup of $\mathcal G$. We also find all the examples where this fails for $\dim(V)\le 6$. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz's recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.