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Coefficient lifting for skew divided differences and triple Schubert coefficients
Abstract
Skew divided difference operators ∂w/v, introduced by Macdonald, play an important role in the study of Schubert polynomials and their structure constants. For permutations u,v,w∈ Sn, we show that the coefficient of each monomial xxα in ∂w/vSSchu(xx;yy), where SSchu(xx;yy) is the double Schubert polynomial, is a single triple Schubert coefficient evaluated at tb=0, whose indices form an explicit lift of (u,v,w) to Sn+absα. As an application, we show that every coefficient of ∂w/vSSchu(xx) is a single Schubert structure constant. This gives a new proof of Kirillov's conjecture that these coefficients are nonnegative.
Related
- Signed puzzles for Schubert coefficients
- Polynomiality of Stretched Schubert Structure Constants and Key Coefficients
- Terminating Zero-Balanced Hypergeometric Series Using Divided Differences
- Long range divided differences, clusters, and Graham-positivity
- Divided differences and complex variations of multiple zeta-star values
BibTeX
@misc{hwang2026coefficient,
title = {{Coefficient lifting for skew divided differences and triple Schubert coefficients}},
author = {Byung-Hak Hwang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21053},
eprint = {2610.21053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21053},
}
RIS
TY - UNPB
TI - Coefficient lifting for skew divided differences and triple Schubert coefficients
AU - Byung-Hak Hwang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21053
UR - https://arxiv.org/abs/2610.21053
AB - Skew divided difference operators ∂_w/v, introduced by Macdonald, play an important role in the study of Schubert polynomials and their structure constants. For permutations u,v,w∈ S_n, we show that the coefficient of each monomial xx^α in ∂_w/vSSch_u(xx;yy), where SSch_u(xx;yy) is the double Schubert polynomial, is a single triple Schubert coefficient evaluated at tb=0, whose indices form an explicit lift of (u,v,w) to S_n+absα. As an application, we show that every coefficient of ∂_w/vSSch_u(xx) is a single Schubert structure constant. This gives a new proof of Kirillov's conjecture that these coefficients are nonnegative.
ER -