Relating Tensor Structures on Representations of General Linear and Symmetric Groups

Abstract

For polynomial representations of GL n of a fixed degree, H. Krause defined a new "internal tensor product" using the language of strict polynomial functors. We show that over an arbitrary commutative base ring k, the Schur functor carries this internal tensor product to the usual Kronecker tensor product of symmetric group representations. This is true even at the level of derived categories. The new tensor product is a substantial enrichment of the Kronecker tensor product. E.g. in modular representation theory it brings in homological phenomena not visible on the symmetric group side. We calculate the internal tensor product over any k in several interesting cases involving classical functors and the Weyl functors. We show an application to the Kronecker problem in characteristic zero when one partition has two rows or is a hook.

Schur Algebras for the Alternating Group and Koszul Duality

“…It follows that η is a natural isomorphism. For the converse, taking N = S, the commutativity of (18) shows that φ is an isomorphism.…”

“…In [17], Krause defined an internal tensor product (⊗) on the category of strict polynomial functors of a fixed degree d. Kulkarni, Srivastava, and Subrahmanyam [18], and independently, Acquilino and Reischuk [1] showed that this internal tensor product, via the Schur functor, is related to the Kronecker tensor product of representations of the symmetric group S d . Krause used this internal tensor product to introduce Koszul duality as the functor (∧…”