Grossberg–Karshon twisted cubes and hesitant walk avoidance
Bibliographic record
Abstract
Let G be a complex semisimple simply connected linear algebraic group.Let λ be a dominant weight for G and Ᏽ = (i 1 , i 2 , . . ., i n ) a word decomposition for an element w = s i 1 s i 2 • • • s i n of the Weyl group of G, where the s i are the simple reflections.In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope associated to λ and Ᏽ, which they called a twisted cube, whose lattice points encode (counted with sign according to a density function) characters of representations of G.In recent work, Harada and Jihyeon Yang proved that the Grossberg-Karshon twisted cube is untwisted (so the support of the density function is a closed convex polytope) precisely when a certain torus-invariant divisor on a toric variety, constructed from the data of λ and Ᏽ, is basepoint-free.This corresponds to the situation in which the Grossberg-Karshon character formula is a true combinatorial formula, in the sense that there are no terms appearing with a minus sign.In this note, we translate this toric-geometric condition to the combinatorics of Ᏽ and λ.More precisely, we introduce the notion of hesitant λ-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when Ᏽ is hesitant-λ-walk-avoiding.Our combinatorial condition imposes strong geometric conditions on the Bott-Samelson variety associated to Ᏽ.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".