POLYNOMIAL FUNCTORS AND TWO-PARAMETER QUANTUM SYMMETRIC PAIRS
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Bibliographic record
Abstract
Abstract We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups GL n , the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair ( $$ {U}_{Q,q}^B $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>U</mml:mi><mml:mrow><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow><mml:mi>B</mml:mi></mml:msubsup></mml:math> ( $$ \mathfrak{gl} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>gl</mml:mi></mml:math> n ), U q ( $$ \mathfrak{gl} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>gl</mml:mi></mml:math> n )) which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra $$ {U}_{Q,q}^B $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>U</mml:mi><mml:mrow><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow><mml:mi>B</mml:mi></mml:msubsup></mml:math> ( $$ \mathfrak{gl} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>gl</mml:mi></mml:math> n ) appears in a Schur–Weyl duality with the type B Hecke algebra $$ {\mathcal{H}}_{Q,q}^B $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow><mml:mi>B</mml:mi></mml:msubsup></mml:math> ( d ). We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it