Filtrations on the representation ring of an affine algebraic group
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Bibliographic record
Abstract
Abstract Let G be an affine algebraic group over a field. The representation ring $\mathrm{R}(G)$ has three standard filtrations, defining the same topology on $\mathrm{R}(G)$: augmentation, Chern and Chow, each of which contained in the next one. For split reductive G, motivated by potential applications related to spin groups, we introduce and study one more filtration, containing the previous ones, which we call induced because it is induced by any of the filtrations on the representation ring of a maximal split torus of G. In the case of semisimple simply connected G, this fourth filtration turns out to be equivalent (in the above topological sense) to the previous three. However, for the spin group $G=\operatorname{\mathrm{Spin}}(d)$ over the complex numbers with $d=7,8$, the new filtration is shown to be strictly larger than the others. It is also shown that for $G=\operatorname{\mathrm{Spin}}(d)$ over an arbitrary field and with any $d\geq7$, the Chern and Chow filtrations on $\mathrm{R}(G)$ are not the same, giving new counter-examples to an extension of Atiyah’s conjecture.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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 it