Inductive systems of the symmetric group, polynomial functors and tensor categories

TL;DR

This paper explores modular representations of symmetric groups via tensor categories, develops a theory of polynomial functors across categories, and links these concepts to the structure of tensor categories and their representations.

Contribution

It introduces a formal framework for polynomial functors in tensor categories and connects this to the classification of symmetric group representations in positive characteristic.

Findings

Classification of polynomial functors relates to symmetric group representations.

Extended strict polynomial functors to arbitrary tensor categories.

Demonstrated connections between tensor category structure and modular representations.

Abstract

We initiate the systematic study of modular representations of symmetric groups that arise via the braiding in (symmetric) tensor categories over fields of positive characteristic. We determine what representations appear for certain examples of tensor categories, develop general principles and demonstrate how this question connects with the ongoing study of the structure theory of tensor categories. We also formalise a theory of polynomial functors as functors that act coherently on all tensor categories. We conclude that the classification of such functors is a different way of posing the above question of which representations of symmetric groups appear. Finally, we extend the classical notion of strict polynomial functors from the category of (super) vector spaces to arbitrary tensor categories, and show that this idea is also a different packaging of the same information.