Symmetric polynomials, generalized Jacobi-Trudi identities and <i>τ</i>-functions
Bibliographic record
Abstract
An element [Φ]∈GrnH+,F of the Grassmannian of n-dimensional subspaces of the Hardy space H+=H2, extended over the field F = C(x1, …, xn), may be associated to any polynomial basis ϕ = {ϕ0, ϕ1, ⋯ } for C(x). The Plücker coordinates Sλ,nϕ(x1,…,xn) of [Φ], labeled by partitions λ, provide an analog of Jacobi’s bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system ϕ to the analog {hi(0)} of the complete symmetric functions generates a doubly infinite matrix hi(j) of symmetric polynomials that determine an element [H]∈Grn(H+,F). This is shown to coincide with [Φ], implying a set of generalized Jacobi identities, extending a result obtained by Sergeev and Veselov [Moscow Math. J. 14, 161–168 (2014)] for the case of orthogonal polynomials. The symmetric polynomials Sλ,nϕ(x1,…,xn) are shown to be KP (Kadomtsev-Petviashvili) τ-functions in terms of the power sums [x] of the xa’s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums ∑λSλ,nϕ([x])Sλ,nθ(t) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".