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We classify all pairs (G,V) with G a closed subgroup in a classical groupwith natural module V over the complex numbers such that G has the samecomposition factors on the kth tensor power of V, for a fixed (small) k. Inparticular, 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 theclassical group. We also find all the examples where this fails for dim(V) < 7.As a consequence of our results, we obtain a short proof of a relatedconjecture of Katz. These conjectures are used in Katz's recent works onmonodromy groups attached to Lefschetz pencils and to character sums overfinite fields. Modular versions of these conjectures are also studied, with aparticular application to random generation in finite groups of Lie type.

Decompositions of small tensor powers and Larsen’s conjecture