An equivalence between truncations of categorified quantum groups and Heisenberg categories
Bibliographic record
Abstract
We introduce a simple diagrammatic 2-category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒜</mml:mi> </mml:math> that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔰𝔩</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> . We show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒜</mml:mi> </mml:math> is equivalent to a truncation of the Khovanov–Lauda categorified quantum group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒰</mml:mi> </mml:math> of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> , and also to a truncation of Khovanov’s Heisenberg 2-category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℋ</mml:mi> </mml:math> . This equivalence is a categorification of the principal realization of the basic representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔰𝔩</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> . As a result of the categorical equivalences described above, certain actions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℋ</mml:mi> </mml:math> induce actions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒰</mml:mi> </mml:math> , and vice versa. In particular, we obtain an explicit action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒰</mml:mi> </mml:math> on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℋ</mml:mi> </mml:math> . The 2-category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒜</mml:mi> </mml:math> can be viewed as a graphical calculus describing the functors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> -induction and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> -restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 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".