SAGBI bases and degeneration of spherical varieties to toric varieties
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Bibliographic record
Abstract
Let X ⊂ P(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope ∆(X). The polytope ∆(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope ∆(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n, C) is shown to hold for other classical groups.
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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 | 0.001 |
| 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) | 0.001 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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