Long range divided differences, clusters, and Graham-positivity
Abstract
We study torus-orbit closures in the type A complete flag variety naturally associated to cones in the positive cluster fan, together with their left Sn-translates. The torus-equivariant degree maps can be computed via composites of long-range divided difference operations encoded by noncrossing alternating forests, and we give combinatorial algorithms to expand the torus-equivariant homology classes into Graham-positive combinations of Schubert cycles. As applications we obtain combinatorial Graham-positive Schubert cycle expansions for all torus-invariant curves (generalizing the AJS-Billey formula for torus-fixed points), generic torus-orbit closures, and Richardson varieties for Bruhat intervals [w,wc'] where c'≤ sn−1sn−2⋯ s1. Projecting to Grassmannians we also obtain Graham-positive Grassmannian Schubert cycle decompositions of torus-orbit closures associated to lattice path matroids on permuted ground sets.
Related
- Coefficient lifting for skew divided differences and triple Schubert coefficients
- Terminating Zero-Balanced Hypergeometric Series Using Divided Differences
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BibTeX
@misc{spink2026long,
title = {{Long range divided differences, clusters, and Graham-positivity}},
author = {Hunter Spink and Vasu Tewari},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10735},
eprint = {2609.10735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10735},
}
RIS
TY - UNPB
TI - Long range divided differences, clusters, and Graham-positivity
AU - Hunter Spink
AU - Vasu Tewari
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10735
UR - https://arxiv.org/abs/2609.10735
AB - We study torus-orbit closures in the type A complete flag variety naturally associated to cones in the positive cluster fan, together with their left S_n-translates. The torus-equivariant degree maps can be computed via composites of long-range divided difference operations encoded by noncrossing alternating forests, and we give combinatorial algorithms to expand the torus-equivariant homology classes into Graham-positive combinations of Schubert cycles. As applications we obtain combinatorial Graham-positive Schubert cycle expansions for all torus-invariant curves (generalizing the AJS-Billey formula for torus-fixed points), generic torus-orbit closures, and Richardson varieties for Bruhat intervals [w,wc'] where c'≤ s_n−1s_n−2⋯ s_1. Projecting to Grassmannians we also obtain Graham-positive Grassmannian Schubert cycle decompositions of torus-orbit closures associated to lattice path matroids on permuted ground sets.
ER -