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

Kevin Coulembier

TL;DR

The paper develops a unified framework to study modular representations of symmetric groups that arise from braiding in positive-characteristic tensor categories. It introduces inductive systems of S_n representations extracted from tensor-category objects and connects them with Frobenius-exactness and Verlinde categories, providing a classification in key examples like Vec, sVec, Ver_p and variants. It then develops two complementary viewpoints—polynomial functors (strict and universal) and their relation to Schur-type representations—to encode the same representation-theoretic information across all tensor categories. The results yield structural insight into when braid actions are surjective, describe maximal ideals in the finitary symmetric group algebra via tensor-category data, and establish deep equivalences between representations of symmetric groups, universal polynomial functors, and strict polynomial functors. The framework paves the way for applications to tensor-category structure theory and cohomological finite-generation questions in modular settings.

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 which 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 representation 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.

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

TL;DR

The paper develops a unified framework to study modular representations of symmetric groups that arise from braiding in positive-characteristic tensor categories. It introduces inductive systems of S_n representations extracted from tensor-category objects and connects them with Frobenius-exactness and Verlinde categories, providing a classification in key examples like Vec, sVec, Ver_p and variants. It then develops two complementary viewpoints—polynomial functors (strict and universal) and their relation to Schur-type representations—to encode the same representation-theoretic information across all tensor categories. The results yield structural insight into when braid actions are surjective, describe maximal ideals in the finitary symmetric group algebra via tensor-category data, and establish deep equivalences between representations of symmetric groups, universal polynomial functors, and strict polynomial functors. The framework paves the way for applications to tensor-category structure theory and cohomological finite-generation questions in modular settings.

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 which 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 representation 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.
Paper Structure (41 sections, 53 theorems, 183 equations)

This paper contains 41 sections, 53 theorems, 183 equations.

Table of Contents

  1. Preliminaries
  2. Conventions and notation
  3. Tensor categories
  4. Deligne tensor product
  5. Module categories and enriched categories
  6. Representations of symmetric groups in tensor categories
  7. Inductive systems of symmetric group representations
  8. Inductive systems
  9. Semisimple inductive systems
  10. Closed inductive systems
  11. Some inclusions between inductive systems
  12. Inductive systems from tensor categories
  13. Representations from the braid action
  14. The proof of Theorem \ref{['ThmTCIS']}
  15. Construction
  16. ...and 26 more sections

Key Result

Lemma 1

For $\Delta:\mathcal{A}\to\mathcal{B}$ in $\mathcal{T}ak$ and $F:\mathcal{C}\to\mathcal{D}$ in $\mathcal{T}ak^{ex}$, the natural isomorphism obtained from the composition isomorphisms to and from $(F\boxtimes \Delta)$, satisfy

Theorems & Definitions (143)

  • Lemma 1
  • Example 1.3.3
  • Example 1.4.3
  • Example 1.4.4
  • Lemma 2
  • proof
  • Definition 2.1.1
  • Remark 3
  • Example 2.1.3
  • Theorem 2.2.2: Kleshchev
  • ...and 133 more