Grossberg–Karshon twisted cubes and hesitant walk avoidance
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Bibliographic record
Abstract
Let $G$ be a complex semisimple simply connected linear algebraic group. Let $\lambda$ be a dominant weight for $G$ and $\mathcal{I} = (i_1, i_2, \ldots, i_n)$ a word decomposition for an element $w = s_{i_1} s_{i_2} \cdots s_{i_n}$ 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 $\lambda$ and $\mathcal{I}$, 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, the first author and Jihyeon Yang prove 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 $\lambda$ and $\mathcal{I}$, 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 $\mathcal{I}$ and $\lambda$. More precisely, we introduce the notion of hesitant $\lambda$-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when $\mathcal{I}$ is hesitant-$\lambda$-walk-avoiding.
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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 | 0.002 | 0.004 |
| 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 | 0.000 |
| 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