Bibliographic record
Abstract
Let G be a finite group. A G-variety X is an algebraic variety with a regular G-action; X is faithful if every 1 6 = g ∈ G acts non-trivially. I will refer to a dominant G-equivariant rational (respectively, regular) map of faithful G-varieties as a rational (respectively, regular) compression. All varieties, actions, vector spaces, maps, etc., are assumed to be defined over a fixed algebraically closed base field k of characteristic zero; all varieties are assumed to be irreducible. I would like to thank V. L. Popov for stimulating discussions and for helpful comments on an earlier draft of this note. 1. Essential dimension Let V be a faithful linear representation of G and let d be the minimal value of dim(X), where the minimum is taken over all rational compressions f: V 99K X. Note that (a) (see [1, Theorem 3.1] or [6, Theorem 3.4(b)]) d depends only on the group G and not on the choice of V, and (b) (cf. [6, Proposition 7.1]) in the definition of d we may assume that X is a G-invariant subvariety of V, i.e., X is the closure of the image of a rational covariant f: V 99K V. The number d is called the essential dimension of G and is usually denoted by ed(G). This number has interesting connections with the algebraic form of Hilbert’s 13th problem, cohomological invariants, generic polynomials and other topics; these connections are described in [1] and [2]. The case where G = Sn is of particular interest. (The notion of essential dimension is also of interest in the context of algebraic groups; see [6] and [7].) Problem 1. Find ed(G) and, in particular, ed(Sn). The value of ed(G) is known if G is an abelian group; see [1, Theorem 6.1]. For symmetric groups, ed(Sn) ≥ [n/2]; this is proved, in different ways,
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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 | 0.002 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".