The monoidal structure on strict polynomial functors

TL;DR

This paper explores the monoidal structure of strict polynomial functors, demonstrating that a key functor to symmetric group representations preserves this structure and linking it to symmetric functions.

Contribution

It proves that the functor from strict polynomial functors to symmetric group representations is monoidal, revealing new structural insights.

Findings

The functor F is monoidal.

Relations between projective functors and permutation modules are established.

Connections to symmetric functions are elucidated.

Abstract

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group which admits a tensor product coming from its Hopf algebra structure. It is classical that there exists a functor F from the category of strict polynomial functors to the category of representations of the symmetric group. Our main result is that this functor F is monoidal. In addition we study the relations under F between projective strict polynomial functors and permutation modules and the link to symmetric functions.