Result 208, Algebra

Finite symmetric tensor categories and the Verlinde tower

Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category Verpn\mathrm{Ver}_{p^n}. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.

Proof

The bigger picture

Why it matters

Finite symmetric tensor categories describe algebraic objects that can be combined, with a consistent rule for exchanging factors. The claimed result places all such systems in positive characteristic within a single tower of mathematical models.

What changes?

The unreviewed manuscript reports that every finite symmetric tensor category over an algebraically closed field k of characteristic p greater than zero admits a fiber functor to a higher Verlinde category at some level n. This is a translation preserving scalar-linear operations, exact sequences, tensor products and their symmetry, without identifying distinct maps. In technical terms, it is k-linear, exact, faithful and strong symmetric monoidal. The theorem includes characteristic two, and the required level n may depend on the category.

What does that help mathematicians do?

The claimed theorem would let researchers study maps and tensor constructions in any category covered by its assumptions through a higher Verlinde category, while retaining their linear and exact-sequence structure. Faithfulness ensures that distinct maps remain distinguishable after translation. This resolves the finite case of the Benson–Etingof–Ostrik conjecture, not the unrestricted conjecture, and does not assert that one fixed level accommodates every finite category.

Are there practical applications?

Its immediate value is foundational: the Verlinde tower becomes a common setting for investigating finite symmetric tensor categories in positive characteristic, including characteristic two. The reported existence theorem supplies a structure-preserving comparison tool. The supplied abstract does not give a procedure for constructing the functor or a bound on the necessary level.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Fiber functors for finite symmetric tensor categories in positive characteristic

September 24, 2026 26 pages

We prove the finite case of the Benson–Etingof–Ostrik conjecture: every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal functor to a finite higher Verlinde category Verpn(k)\mathop{\mathrm{Ver}}\nolimits _{p^n}(k). The result includes characteristic two, and the level n may depend on the category.

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Cite (BibTeX) 
@misc{OAI:Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026,
  author = {{OpenAI}},
  title = {{Fiber functors for finite symmetric tensor categories in positive characteristic}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf}{OAI:Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026}},
  year = {2026}
}

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