Bibliographic record
Abstract
One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GL n = m , i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m.In this paper we study the essential dimension of groups of the formwhere C is a central subgroup of GL n 1 GL n r .Equivalently, we are interested in the essential dimension of a generic r-tuple .A 1 ; : : : ; A r / of central simple algebras such that deg.A i / D n i and the Brauer classes of A 1 ; : : : ; A r satisfy a system of homogeneous linear equations in the Brauer group.The equations depend on the choice of C via the error-correcting code Code.C / which we naturally associate to C .We focus on the case where n 1 ; : : : ; n r are powers of the same prime.The upper and lower bounds on ed.G / we obtain are expressed in terms of coding-theoretic parameters of Code.C /, such as its weight distribution.Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r 3; in some cases we even compute the exact value.The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form .GL n 1 GL n r /=C .This description and its corollaries are used throughout the paper.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".