The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair (SO<sub><i>p</i>+<i>q</i></sub>, SO<i><sub>p</sub></i> × SO<i><sub>q</sub></i>)
Bibliographic record
Abstract
Abstract The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group K = O p ( C ) × O q ( C ) acts on the space M p, q of p × q complex matrices by ( a ; b ) · x = axb –1 , and so does its identity component K 0 = SO p ( C )×SO q ( C ). A K -orbit (or K 0 -orbit) in M p,q is said to be nilpotent if its closure contains the zero matrix. The closure, , of a nilpotent K -orbit (resp. K 0 -orbit) in M p,q is a union of and some nilpotent K -orbits (resp. K 0 -orbits) of smaller dimensions. The description of the closure of nilpotent K -orbits has been known for some time, but not so for the nilpotent K 0 -orbits. A conjecture describing the closure of nilpotent K 0 -orbits was proposed in [11] and verièd when min( p , q ) ≤ 7. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to and determination of the basic relative invariants of these spaces. The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra so( p , q ) under the adjoint action of the identity component of the real orthogonal group O( p , q ).
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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 | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".