Criteria for rational smoothness of some symmetric orbit closures
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Bibliographic record
Abstract
Let $G$ be a connected reductive linear algebraic group over $ℂ$ with an involution $θ$ . Denote by $K$ the subgroup of fixed points. In certain cases, the $K-orbits$ in the flag variety $G/B$ are indexed by the twisted identities $ι (θ ) = {θ (w^{-1})w | w∈W}$ in the Weyl group $W$. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph'' whose vertices form a subset of $ι (θ )$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $ι (θ )$ is rank symmetric. In the special case $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one'', needs to be examined. This generalises a result of Deodhar for type A Schubert varieties. Soit $G$ un groupe algébrique connexe réductif sur $ℂ$, équipé d'une involution $θ$ . Soit $K$ le sousgroupe de ses points fixes. Dans certains cas, les orbites des points de la variété de drapeaux $G/B$ sous l'action de $K$ sont indexées par les identités tordues, $ι (θ ) = {θ (w^{-1})w | w∈W}$, du groupe de Weyl $W$. Sous cette hypothèse, on établit un critère pour la lissité rationnelle des adhérences des orbites, qui généralise des résultats classiques de Carrell et Peterson pour les variétés de Schubert. Plus précisément, on peut déterminer si l'adhérence d'une orbite est rationnellement lisse en examinant les degrés dans un "Graphe de Bruhat" dont les sommets forment un sous-ensemble de $ι (θ )$. En outre, l'adhérence d'une orbite est partout rationnellement lisse si et seulement si l'intervalle correspondant dans l'ordre de Bruhat de $ι (θ )$ est symétrique respectivement au rang. Dans le cas particulier $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, nous améliorons notre critère en montrant qu'il suffit d'examiner le degré d'un seul sommet, celui "du bas". Ceci généralise un résultat de Deodhar pour les variétés de Schubert de type A.
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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 | 0.002 |
| 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 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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