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

We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to obtain a new proof of the Steinberg tensor product theorem, as well as new generalizations and variants of it. Our connectedness bounds depend on numerical invariants which refine the notion of projective and injective dimensions. These numerical invariants seem also relevant to other problems, for example to study the cohomological behavior of the Schur functor.