It is known for a long time that polynomial representations of GL_n(k) stabilize when n grows, i.e. Schur algebras S(n, d) are all Morita equivalent when n ≥ d. A model of the category of stable polynomial representations is given by the strict polynomial functors of Friedlander and Suslin. Using the formalism of strict polynomial functors, we prove a rather counter-intuitive results on cup products, namely that the cup product

Ext^∗(M, N) ⊗ Ext^∗(P^(r), Q^(r)) → Ext^∗(M ⊗ P^(r), N ⊗ Q^(r))

induces an isomorphism in low degrees when M, N, P, Q are stable polynomial representations. We shall explain some consequences of these results (including a new proof of the Steinberg tensor product theorem, as well as more general structure theorems which generalize it) and connections with the cohomology of the symmetric group.