On the zero set of <i>G</i>-equivariant maps
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Bibliographic record
Abstract
Abstract Let G be a finite group acting on vector spaces V and W and consider a smooth G -equivariant mapping f : V → W . This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G . It is known from results of Bierstone and Field on G -transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W . We define an index s (Σ) for isotropy subgroups Σ of G which is the difference of the dimension of the fixed point subspace of Σ in V and W . Our main result states that if V contains a subspace G -isomorphic to W , then for every maximal isotropy subgroup Σ satisfying s (Σ) > s ( G ), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup Σ of dimension s (Σ). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G -reversible equivariant vector fields.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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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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