The closure ordering of adjoint nilpotent orbits in so(p,q)
Bibliographic record
Abstract
Let ${\mathcal{O}}$ be a nilpotent orbit in ${\mathfrak{so}}(p,q)$ under the adjoint action of the full orthogonal group ${\rm{O}}(p,q)$. Then the closure of ${\mathcal{O}}$ (with respect to the Euclidean topology) is a union of ${\mathcal{O}}$ and some nilpotent ${\rm{O}}(p,q)$-orbits of smaller dimensions. In an earlier work, the first author has determined which nilpotent ${\rm{O}}(p,q)$-orbits belong to this closure. The same problem for the action of the identity component ${\rm{SO}}(p,q)^0$ of ${\rm{O}}(p,q)$ on ${\mathfrak{so}}(p,q)$ is much harder and we propose a conjecture describing the closures of the nilpotent ${\rm{SO}}(p,q)^0$-orbits. The conjecture is proved when $\min(p,q)\le7$. Our method is indirect because we use the Kostant-Sekiguchi correspondence to translate the problem to that of describing the closures of the unstable orbits for the action of the complex group ${\rm{SO}} p({\bf{C}})\times{\rm{SO}} q({\bf{C}})$ on the space $M_{p,q}$ of complex $p\times q$ matrices with the action given by $(a,b)\cdot x=axb^<-1>$. The fact that the Kostant--Sekiguchi correspondence preserves the closure relation has been proved recently by Barbasch and Sepanski.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".