It is known for a long time that polynomial representations of GLn(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.
This video was produced by Syracuse University Department of Mathematics as part of ICRA 2016.
