Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one
Why this work is in the frame
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Bibliographic record
Abstract
Let [math] and let [math] and [math] be two line bundles on [math] . Consider the cup-product map ¶\n<math display="block">\n<mrow> <msup><mrow><mo class="qopname">H</mo></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub> </mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mo class="qopname">X</mo><mo class="MathClass-punc">,</mo><msub><mrow><mo class="qopname">L</mo></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-bin">⊗</mo><msup><mrow><mo class="qopname"> H</mo></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub> </mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mo class="qopname">X</mo><mo class="MathClass-punc">,</mo><msub><mrow><mo class="qopname">L</mo></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow><munderover> <mo class="MathClass-rel">→</mo><mrow/><mrow><mo class="MathClass-bin">∪</mo></mrow></munderover><msup><mrow><mo class="qopname">H</mo></mrow><mrow><mi>d</mi></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mo class="qopname">X</mo><mo class="MathClass-punc">,</mo><mo class="qopname">L</mo></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-punc">,</mo> </mrow>\n</math>\n¶ where [math] and [math] . We answer two natural questions about the map above: When is it a nonzero homomorphism of representations of [math] ? Conversely, given generic irreducible representations [math] and [math] , which irreducible components of [math] may appear in the right hand side of the equation above? For the first question we find a combinatorial condition expressed in terms of inversion sets of Weyl group elements. The answer to the second question is especially elegant: the representations [math] appearing in the right hand side of the equation above are exactly the generalized PRV components of [math] of stable multiplicity one. Furthermore, the highest weights [math] corresponding to the representations [math] fill up the generic faces of the Littlewood–Richardson cone of [math] of codimension equal to the rank of [math] . In particular, we conclude that the corresponding Littlewood–Richardson coefficients equal one.
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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 | 0.001 | 0.002 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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