Homogeneous strict polynomial functors as unstable modules

TL;DR

This paper explores the relationship between strict polynomial functors and unstable modules, demonstrating that a functor linking Schur algebras and Steenrod algebra is fully faithful when restricted to a specific subcategory.

Contribution

It proves that the restriction of Hai's functor to strict polynomial functors of a fixed degree is fully faithful, clarifying the structure of this relationship.

Findings

Hai's functor is fully faithful on strict polynomial functors of a fixed degree

Establishes a precise connection between Schur algebras and Steenrod algebra

Enhances understanding of polynomial functors in algebraic topology

Abstract

A relation between Schur algebras and Steenrod algebra is shown in [Hai10] where to each strict polynomial functor the author associates an unstable module. We show that the restriction of Hai's functor to the subcategory of strict polynomial functors of a given degree is fully faithfull.