The Ariki–Koike algebras and Rogers–Ramanujan type partitions
Bibliographic record
Abstract
Abstract Ariki and Mathas (Math Z 233(3):601–623, 2000) showed that the simple modules of the Ariki–Koike algebras $$\mathcal {H}_{\mathbb {C},v;Q_1,\ldots , Q_m}\big (G(m, 1, n)\big )$$ H C , v ; Q 1 , … , Q m ( G ( m , 1 , n ) ) (when the parameters are roots of unity and $$v \ne 1$$ v ≠ 1 ) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit their generating function relation for the $$v=-1$$ v = - 1 case. In particular, this $$v=-1$$ v = - 1 scenario is of special interest as the corresponding Kleshchev multipartitions are closely tied with generalized Rogers–Ramanujan type partitions when $$Q_1=\cdots =Q_a=-1$$ Q 1 = ⋯ = Q a = - 1 and $$Q_{a+1}=\cdots =Q_m =1$$ Q a + 1 = ⋯ = Q m = 1 . Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for the above choice of parameters. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block, which are, as widely known, labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities for the $$m=2$$ m = 2 scenario.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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".