Invariant algebras and major indices for classical Weyl groups
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Bibliographic record
Abstract
Given a classical Weyl group W, that is, a Weyl group of type A, B or D, one can associate with it a polynomial with integral coefficients ZW given by the ratio of the Hilbert series of the invariant algebras of the natural action of W and Wt on the ring of polynomials C[x1,…,xn]⊗t. We introduce and study several statistics on the classical Weyl groups of type B and D and show that they can be used to give an explicit formula for ZDn. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, Dmaj and ned on Dn. The statistic Dmaj, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for Bn; namely, it allows us to find an explicit formula for ZDn. Our proof is based on the theory of t-partite partitions introduced by Gordon and further studied by Garsia and Gessel. Using similar ideas, we define the Mahonian statistic ned also on Bn and we find a new and simpler proof of the Adin–Roichman formula for ZBn. Finally, we define a new descent number Ddes on Dn so that the pair (Ddes,Dmaj) gives a generalization to Dn of the Carlitz identity on the Eulerian–Mahonian distribution of descent number and major index on the symmetric group. 2000 Mathematics Subject Classification 05E15 (primary), 05A19 (secondary).
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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