A đ-deformation of an algebra of Klyachko, and Macdonaldâs reduced word formula
Bibliographic record
Abstract
There is a striking similarity between Macdonaldâs reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety P e r m n \mathrm {Perm}_n as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a q q -deformation of the S n \mathbb {S}_n -invariant part of the rational cohomology ring of P e r m n \mathrm {Perm}_n , which we call the q q -Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows. A q q -analogue of divided symmetrization ( q q -DS) using YangâBaxter elements in the Hecke algebra. It is a linear form that picks up coefficients in the squarefree basis. A relation between q q -DS and the ideal of quasisymmetric polynomials involving work of AvalâBergeronâBergeron. A family of polynomials in q q with nonnegative integral coefficients that specialize to Postnikovâs mixed Eulerian numbers when q = 1 q=1 . We refer to these new polynomials as remixed Eulerian numbers. For q > 0 q>0 , their normalized versions occur as probabilities in the internal diffusion limited aggregation stochastic process. A lift of Macdonaldâs reduced word identity in the q q -Klyachko algebra. The Schubert expansion of the Chow class of the standard split DeligneâLusztig variety in type A A , when q q is a prime power.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.012 | 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".