Essential dimension of representations of algebras
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Bibliographic record
Abstract
Let k be a field, A be a finitely generated associative k -algebra and Rep _A[n] be the functor Fields _k\to Sets, which sends a field K containing k to the set of isomorphism classes of representations of A_K of dimension at most n . We study the asymptotic behavior of the essential dimension of this functor, i.e., the function r_A(n) := \mathrm {ed}_k (Rep _A[n]) , as n\to\infty . In particular, we show that the rate of growth of r_A(n) determines the representation type of A . That is, r_A(n) is bounded from above if A is of finite representation type, grows linearly if A is of tame representation type, and grows quadratically if A is of wild representation type. Moreover, r_A(n) allows us to construct invariants of algebras which are finer than the representation type.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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