The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials

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In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials fᵢ,ⱼ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial fᵢ,ⱼ is an alternating sum of certain Schubert polynomials.