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We find conditions such that cup products induce isomorphisms in low degreesfor extensions between stable polynomial representations of the general lineargroup. We apply this result to prove generalizations and variants of theSteinberg tensor product theorem. Our connectedness bounds for cup product mapsdepend on numerical invariants which seem also relevant to other problems, forexample to study the cohomological behavior of the Schur functor.

Connectedness of cup products for polynomial representations of $GL_n$ and applications