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

Kevin Coulembier, Serina Hu

TL;DR

This work analyzes polynomial functors in the incompressible tensor category $Ver_4^+$ in characteristic $2$, framing them as (super) polynomial functors and developing a systematic framework to classify additive, exact, and simple polynomial functors. It combines the representation theory of general linear groups inside $Ver_4^+$ with inductive and reflective techniques (Frobenius twists and odd reflections) to describe how simple polynomial functors evaluate on objects and to identify when objects are $d$-discerning or $d$-faithful. The paper provides a complete classification of simple polynomial functors via combinatorial data $(\lambda|\mu)$, and a precise account of additive and exact polynomial functors, including a five-item indecomposable structure for degrees $d=2^\ell$ ($\ell\ge 2$). It also develops Frobenius twist behavior and an odd-reflection calculus in characteristic two, informing how polynomial functors interact with $Ver_4^+$ and its GL-representations, with detailed consequences for invariant theory and related finite-generation questions. Overall, the results advance understanding of polynomial functors in incompressible tensor categories and illuminate the structure of simple and additive polynomial representations in $Ver_4^+$, with potential applications to cohomology generation and higher Frobenius functors.

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.

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

TL;DR

This work analyzes polynomial functors in the incompressible tensor category in characteristic , framing them as (super) polynomial functors and developing a systematic framework to classify additive, exact, and simple polynomial functors. It combines the representation theory of general linear groups inside with inductive and reflective techniques (Frobenius twists and odd reflections) to describe how simple polynomial functors evaluate on objects and to identify when objects are -discerning or -faithful. The paper provides a complete classification of simple polynomial functors via combinatorial data , and a precise account of additive and exact polynomial functors, including a five-item indecomposable structure for degrees (). It also develops Frobenius twist behavior and an odd-reflection calculus in characteristic two, informing how polynomial functors interact with and its GL-representations, with detailed consequences for invariant theory and related finite-generation questions. Overall, the results advance understanding of polynomial functors in incompressible tensor categories and illuminate the structure of simple and additive polynomial representations in , with potential applications to cohomology generation and higher Frobenius functors.

Abstract

We study polynomial functors in the incompressible category , 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.
Paper Structure (16 sections, 34 theorems, 70 equations)

This paper contains 16 sections, 34 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor categories
  4. The higher Verlinde category $\mathop{\mathrm{Ver}}\nolimits_4^+$
  5. General linear groups in $\mathop{\mathrm{Ver}}\nolimits_4^+$
  6. Generalities on polynomial functors
  7. Closed inductive systems
  8. Polynomial functors
  9. Additive polynomial functors
  10. Example: super vector spaces
  11. Polynomial Functors in $\mathop{\mathrm{Ver}}\nolimits_4^+$
  12. A Frobenius twist
  13. Odd reflections in characteristic two
  14. Simple polynomial functors
  15. Additive polynomial functors
  16. ...and 1 more sections

Key Result

Proposition 2.1

The fundamental group $\pi_1$ of $\mathop{\mathrm{Ver}}\nolimits_4^+$ has coordinate algebra $k[\pi_1] = H.$ An element $b \in \pi_1(A)\subset A$ acts as the identity on $A \otimes \mathds{1}$ and as $$ on $A \otimes P$.

Theorems & Definitions (96)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 3.1
  • Proposition 3.2
  • proof
  • ...and 86 more