The principal indecomposable modules of the dilute Temperley-Lieb algebra
Bibliographic record
Abstract
The Temperley-Lieb algebra \documentclass[12pt]{minimal}\begin{document}$\mathsf {TL}_{n}(\beta )$\end{document}TLn(β) can be defined as the set of rectangular diagrams with n points on each of their vertical sides, with all points joined pairwise by non-intersecting strings. The multiplication is then the concatenation of diagrams. The dilute Temperley-Lieb algebra \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}(\beta )$\end{document}dTLn(β) has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like \documentclass[12pt]{minimal}\begin{document}$\mathsf {TL}_{n}$\end{document}TLn, the dilute \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn depends on a parameter \documentclass[12pt]{minimal}\begin{document}$\beta \in \mathbb {C}$\end{document}β∈C, often given as β = q + q−1 for some \documentclass[12pt]{minimal}\begin{document}$q\in \mathbb {C}^\times$\end{document}q∈C×. In statistical physics, the algebra plays a central role in the study of dilute loop models. The paper is devoted to the construction of its principal indecomposable modules. Basic definitions and properties are first given: the dimension of \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn, its break up into even and odd subalgebras and its filtration through n + 1 ideals. The standard modules \documentclass[12pt]{minimal}\begin{document}$\mathsf {S}_{n,k}$\end{document}Sn,k are then introduced and their behaviour under restriction and induction is described. A bilinear form, the Gram product, is used to identify their (unique) maximal submodule \documentclass[12pt]{minimal}\begin{document}$\mathsf {R}_{n,k}$\end{document}Rn,k which is then shown to be irreducible or trivial. It is then noted that \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn is a cellular algebra. This fact allows for the identification of complete sets of non-isomorphic irreducible modules and projective indecomposable ones. The structure of \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn as a left module over itself is then given for all values of the parameter q, that is, for both q generic and a root of unity.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.004 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".