Séminaire Lotharingien de Combinatoire, 95B.37 (2026), 11 pp.

Yibo Gao and Rui Xiong

Positivity in Triple Schubert Calculus

Abstract. We prove Samuel's conjecture on certain Graham positivity of the expansion coefficient of two double Schubert polynomials in three sets of variables by establishing a refined version of Graham's positivity theorem. As a corollary, we prove Kirillov's conjecture on the positivity of skew divided difference operators applied to Schubert polynomials.

Received: November 15, 2025. Accepted: February 15, 2026. Final version: April 1, 2026.

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