Bibliographic record
Abstract
Hecke algebras associated to reductive groups over a finite field Fq were introduced in order to decompose representations of those groups induced from parabolic subgroups. They have subsequently become ubiq uitous in representation theory, but often as algebras whose coefficients are polynomials, in which variables replace various powers of q. The existence of these Hecke algebras with polynomial coefficients is not quite trivial. There are essentially two constructions in the literature. One originates in Exercices IV.2225 of [Bourbaki:1968], and is appar ently due originally to Jacques Tits. There are other accounts patterned after this argument, for example in [Humphreys:1990] and [Carter:1993]. My reaction to these is that they are clever but obscurely motivated— several tools used in the proof do not occur subsequently in the theory. There is a rather different, proof in [Eriksson:1994], which has much to be said for it. In this paper I offer a third, having something in common with each of these, but with what I consider to be a more direct approach. It was originally suggested in the course of writing programs for dealing with Hecke algebras. I intend this paper to be largely selfcontained, readable by novices. Before I present the proof of the general theorem, I recall the origins of the main theorem by looking at what happens for Hecke algebras of reductive groups defined over finite fields. Similar discussions are not difficult to find in the literature, but they are
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 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 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".