<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi mathvariant="normal">Sym</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -modules and Nabla of hook-indexed Schur functions
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Bibliographic record
Abstract
The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> ) of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> variables, both as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> -modules and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="normal">Sym</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -modules. We construct explicit such modules associated to any hook shape partitions. For the two sets of variables case, we conjecture that the associated graded Frobenius characteristic corresponds to the effect of the operator Nabla on the corresponding hook-indexed Schur function, up to a usual renormalization. We prove identities that give indirect support to this conjecture, and show that its restriction to one set of variables holds. We further give indications on how the several sets context gives a better understanding of questions regarding the structures of these modules and the links between them.
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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 | 0.007 | 0.007 |
| Meta-epidemiology (narrow) | 0.003 | 0.007 |
| Meta-epidemiology (broad) | 0.002 | 0.005 |
| Bibliometrics | 0.002 | 0.006 |
| Science and technology studies | 0.007 | 0.004 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.007 | 0.009 |
| Research integrity | 0.006 | 0.007 |
| Insufficient payload (model declined to judge) | 0.166 | 0.004 |
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