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

TL;DR

This paper establishes conditions under which cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of GL_n, with applications to generalizing the Steinberg tensor product theorem.

Contribution

It provides new criteria for cup product isomorphisms and applies them to extend classical results like the Steinberg tensor product theorem.

Findings

Cup product maps induce isomorphisms under specific conditions.

Connectedness bounds depend on numerical invariants.

Results have implications for the cohomology of Schur functors.

Abstract

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 prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, for example to study the cohomological behavior of the Schur functor.