Bibliographic record
Abstract
Abstract The q-Whittaker function $$W_\lambda (\textbf{x};q)$$ W λ ( x ; q ) associated to a partition $$\lambda $$ λ is a q-analogue of the Schur function $$s_\lambda (\textbf{x})$$ s λ ( x ) , and is defined as the $$t=0$$ t = 0 specialization of the Macdonald polynomial $$P_\lambda (\textbf{x};q,t)$$ P λ ( x ; q , t ) . We show combinatorially how to expand $$W_\lambda (\textbf{x};q)$$ W λ ( x ; q ) in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q. This yields an expression analogous to a well-known formula for the Hall–Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, proving the Cauchy identity for q-Whittaker functions. We call our probabilistic bijection the q-Burge correspondence, and prove that in the limit as $$q\rightarrow 0$$ q → 0 , we recover a description of the classical Burge correspondence (also known as column RSK) due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary double coset of $$\text {GL}_n$$ GL n modulo two parabolic subgroups, which we find to be of independent interest. As an application, we use the q-Burge correspondence to count isomorphism classes of certain modules over the preprojective algebra of a type A quiver (i.e. a path), refined according to their socle filtrations. This develops a connection between the combinatorics of symmetric functions and the representation theory of preprojective algebras.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.019 | 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 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".