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Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six

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Zhengwei Liu, Sebastien Palcoux, Yunxiang Ren

Category Theory · Submitted 20 Oct 2020

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six.

how it is stated · asserted 1.00 read the record →

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arXiv ID2010.10264 (version 5)
Pith IDIT6ZFZMT, Pith's permanent record of this paper and its review
CitationSebastien Palcoux, Yunxiang Ren, Zhengwei Liu. "Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six." arXiv:2010.10264 (2020). https://pith.science/paper/IT6ZFZMT
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,509 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. contribution 0.95 0.92 0.62 asserted 1.00
s1 Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. result 0.73 0.90 0.07 asserted 1.00
s2 Each of the remaining 47 fusion rings admits a unitary complex categorification. result 1.00 0.77 0.24 asserted 1.00
s3 We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation. result 0.63 0.96 0.16 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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  • references and citing sentences
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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.

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References
How the literature uses it

9 arXiv papers cite this in 16 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 10 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 1 paper
  • To rule out general categorifiability for fusion rings the zero-spectrum criterion (ZSC) from [ref] (and also [ref] ) can be used: For a fusion ring $R$, if there are indices $i_j, 1 ≤j ≤9,$ such that $N_i_4i_1^i_6$, $N_i_5 i_4^i_2$, $N_i_5 i_6^i_3$, $N_i_7 i_9^i_1$ , $N_i_2 i_7^i_8$, $N_i_8 i_9^i_3$ are non-zero, and [equation] then $R$ cannot be categorified.

    On Low Rank Fusion Rings · Categorifiability · 0.92

Compares with it 1 paper
Cites it as background 2 papers

What it cites · 4 sentences without a confident label

Uses the method or tool of 2 papers
  • Here is the commutative Schur product criterion [ref] : If there is a triple $(j_1,j_2,j_3)$ such that [equation] then $ℱ$ admits no unitary categorification.

    Fusion Bialgebras and Fourier Analysis · Schur product criterion · 0.86

  • Rowell, the result follows from the new zesting construction [ref] using the braiding of $(N)_2$ and its $C_2$-grading, which twists the associativity by a $3$-cocycle.

    Braided zesting and its applications · Models by zesting construction · 0.94

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