Coefficient lifting for skew divided differences and triple Schubert coefficients

Byung-Hak Hwang

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arXiv · 2610.21053

Coefficient lifting for skew divided differences and triple Schubert coefficients

Byung-Hak Hwang

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Abstract

Skew divided difference operators $\partial_{w/v}$, introduced by Macdonald, play an important role in the study of Schubert polynomials and their structure constants. For permutations $u,v,w\in S_n$, we show that the coefficient of each monomial $\xx^α$ in $\partial_{w/v}\SSch_u(\xx;\yy)$, 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 $\partial_{w/v}\SSch_u(\xx)$ is a single Schubert structure constant. This gives a new proof of Kirillov's conjecture that these coefficients are nonnegative.

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Byung-Hak Hwang. 2026-10-15. Coefficient lifting for skew divided differences and triple Schubert coefficients. https://arxiv.org/abs/2610.21053

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