Fusion and monodromy in the Temperley-Lieb category
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Bibliographic record
Abstract
Graham and Lehrer (1998) introduced a Temperley-Lieb category \ctl whose objects are the non-negative integers and the morphisms in \Hom(n,m) are the link diagrams from n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> to m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>m</mml:mi> </mml:math> nodes. The Temperley-Lieb algebra \tl n is identified with \Hom(n,n) . The category \ctl is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on \ctl . We introduce a module category \modtl whose objects are functors from \ctl to \mathsf{Vect}_{\mathbb C} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖵</mml:mi> <mml:mi>𝖾</mml:mi> <mml:mi>𝖼</mml:mi> <mml:mi>𝗍</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℂ</mml:mi> </mml:mstyle> </mml:msub> </mml:math> and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for \ctl to induce the structure of a ribbon category on \modtl(\beta=-q-q^{-1}) , when q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>q</mml:mi> </mml:math> is not a root of unity. We discuss how the braiding on \ctl and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it