Capelli operators for spherical superharmonics and the Dougall-Ramanujan\n identity
Bibliographic record
Abstract
Let $(V,\\omega)$ be an orthosympectic $\\mathbb Z_2$-graded vector space and\nlet $\\mathfrak g:=\\mathfrak{gosp}(V,\\omega)$ denote the Lie superalgebra of\nsimilitudes of $(V,\\omega)$. When the space $\\mathscr P(V)$ of superpolynomials\non $V$ is \\emph{not} a completely reducible $\\mathfrak g$-module, we construct\na natural basis $D_\\lambda$ of Capelli operators for the algebra of $\\mathfrak\ng$-invariant superpolynomial superdifferential operators on $V$, where the\nindex set $\\mathcal P$ is the set of integer partitions of length at most two.\nWe compute the action of the operators $D_\\lambda$ on maximal indecomposable\ncomponents of $\\mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation\npolynomials. Our results show that, unlike the cases where $\\mathscr P(V)$ is\ncompletely reducible, the eigenvalues of a subfamily of the $D_\\lambda$ are\n\\emph{not} given by specializing the Knop-Sahi polynomials. Rather, the\nformulas for these eigenvalues involve suitably regularized forms of these\npolynomials. In addition, we demonstrate a close relationship between our\neigenvalue formulas for this subfamily of Capelli operators and the\nDougall-Ramanujan hypergeometric identity.\n We also transcend our results on the eigenvalues of Capelli operators to the\nDeligne category $\\mathsf{Rep}(O_t)$. More precisely, we define categorical\nCapelli operators $\\{\\mathbf D_{t,\\lambda}\\}_{\\lambda\\in\\mathcal P}^{}$ that\ninduce morphisms of indecomposable components of symmetric powers of $\\mathsf\nV_t$, where $\\mathsf V_t$ is the generating object of $\\mathsf{Rep}(O_t)$. We\nobtain formulas for the eigenvalue polynomials associated to the\n$\\left\\{\\mathbf D_{t,\\lambda}\\right\\}_{\\lambda\\in\\mathcal P}$ that are\nanalogous to our results for the operators $\\{D_\\lambda\\}_{\\lambda\\in\\mathcal\nP}^{}$.\n
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| 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) | 0.000 | 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".