Rank 2 affine MV polytopes
Bibliographic record
Abstract
We give a realization of the crystal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis negative normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B(-\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal s normal l With caret Subscript 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\widehat {\mathrm {sl}}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis negative normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B(-\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , in terms of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> involution. The polygons we use have combinatorial properties suggesting they are the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal s normal l With caret Subscript 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\widehat {\mathrm {sl}}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara’s similarity of crystals we also give MV polytopes for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 2 Superscript left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">A_2^{(2)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the other rank 2 affine Kac-Moody algebra.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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".