Polynomial functors in $\text{Ver}_4^+$

TL;DR

This paper investigates polynomial functors within the category Ver_4^+, classifying simple functors, analyzing their evaluations, and exploring their action on objects, especially in the context of characteristic 2.

Contribution

It provides a classification of additive, exact, and simple polynomial functors in Ver_4^+ and describes their evaluations and faithfulness properties.

Findings

Classified simple polynomial functors in Ver_4^+.

Determined objects not annihilated by polynomial functors of a given degree.

Identified objects where the symmetric group algebra acts faithfully.

Abstract

We study polynomial functors in the incompressible category $\text{Ver}_4^+$, which can be viewed as super polynomial functors in characteristic 2. Concretely, we classify additive, exact and simple polynomial functors, and describe how simple polynomial functors evaluate on arbitrary objects. We also determine which objects are not annihilated by any polynomial functors of a given degree and for which objects the symmetric group algebra acts faithfully via the braiding.