A Characterization of the Quantum Cohomology Ring of <i>G/B</i> and Applications
Bibliographic record
Abstract
Abstract We observe that the small quantum product of the generalized flag manifold G/B is a product operation ★ on H*(G/B) ⊗ ℝ[ q 1 , . . . , q l ] uniquely determined by the facts that it is a deformation of the cup product on H*(G/B) ; it is commutative, associative, and graded with respect to deg( qi ) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring ( H *( G/B )⊗ℝ[ q 1 , . . . , q l ], ★) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for G/B : the quantumChevalley formula of D. Peterson (see also Fulton andWoodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for G = SL ( n, ℂ ). The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum -module of G/B one can decode all information about the quantum cohomology of this space.
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How this classification was reachedexpand
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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".