A bijective proof of a factorization formula for Macdonald polynomials at roots of unity
Bibliographic record
Abstract
We give a combinatorial proof of the factorization formula of modified Macdonald polynomials $\widetilde{H}_{\lambda} (X;q,t)$ when $t$ is specialized at a primitive root of unity. Our proof is restricted to the special case where $\lambda$ is a two columns partition. We mainly use the combinatorial interpretation of Haiman, Haglund and Loehr giving the expansion of $\widetilde{H}_{\lambda} (X;q,t)$ on the monomial basis. Nous présentons une preuve combinatoire de la formule de factorisation des polynômes de Macdonald modifiés $\widetilde{H}_{\lambda} (X;q,t)$ quand $t$ est spécialisé à une racine primitive de l'unité. Notre preuve se restreint au cas particulier des partitions $\lambda$ n'ayant que deux colonnes. On utilise principalement l'interprétation combinatoire de Haglund, Haiman and Loehr donnant le développement de $\widetilde{H}_{\lambda} (X;q,t)$ sur la base des fonctions monomiales.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.018 | 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".