Symmetric association schemes arising from abstract regular polytopes
Bibliographic record
Abstract
This article investigates the question of when every double coset of a string $C$-group $G$ relative to its vertex stabilizer subgroup $H$ is represented by an involution. We show that this is the case for every finite string Coxeter group except in the $\{5,3,3\}$ case of type $H_4$, and for the infinite Coxeter groups of Schläfli type $\{4,4\}$ and $\{3,6\}$. From this it is immediate that, for every string $C$-group of these types, the double coset algebra $\mathbb{C}[G/\!\!/H]$ is commutative and all of its characters are realizable over $\mathbb{R}$. In particular, the abstract regular polytopes with these automorphism groups have a polyhedral realization cone.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 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".