  We classify all pairs (G,V) with G a closed subgroup in a classical group
with natural module V over the complex numbers such that G has the same
composition factors on the kth tensor power of V, for a fixed (small) k. 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 the
classical 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 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.
