Invariants for the modular cyclic group of prime order via classical invariant theory
Bibliographic record
Abstract
Let \mathbb F be any field of characteristic p . It is well-known that there are exactly p inequivalent indecomposable representations V_1,V_2,\dots,V_p of C_p defined over \mathbb F . Thus if V is any finite dimensional C_p -representation there are non-negative integers 0\leq n_1,n_2,\dots, n_k \leq p-1 such that V \cong \oplus_{i=1}^k V_{n_i+1} . It is also well-known there is a unique (up to equivalence) d+1 dimensional irreducible complex representation of SL _2(\mathbb C) given by its action on the space R_d of d forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of C_p -invariants \mathbb F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p} to the computation of the classical ring of invariants (or covariants) \mathbb C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\mathrm {SL}_2(\mathbb C)} . This shows that the problem of computing modular C_p invariants is equivalent to the problem of computing classical SL _2(\mathbb C) invariants. This allows us to compute for the first time the ring of invariants for many representations of C_p . In particular, we easily obtain from this generators for the rings of vector invariants \mathbb F[m\,V_2]^{C_p} , \mathbb F[m\,V_3]^{C_p} and \mathbb F[m\,V_4]^{C_p} for all m \in \mathbb N . This is the first computation of the latter two families of rings of invariants.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".