We define and study the rational Schubert, rational Grothendieck, rational key polynomials in an effort to understand Molev’s dual Schur functions from the viewpoint of Lascoux. | Turk J Math (2015) 39: 439 – 452 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Rational Schubert polynomials 1 K¨ ur¸sat AKER1 , Nesrin TUTAS ¸ 2,∗ Department of Mathematics, Middle East Technical University, Northern Cyprus Campus, Kalkanlı, G¨ uzelyurt, Mersin 10, Turkey 2 Department of Mathematics, Akdeniz University, Antalya, Turkey Received: • Accepted/Published Online: • Printed: Abstract: We define and study the rational Schubert, rational Grothendieck, rational key polynomials in an effort to understand Molev’s dual Schur functions from the viewpoint of Lascoux. Key words: Rational Schubert polynomials, Schubert calculus 1. Introduction In this work, we introduce a new set of combinatorially defined nonsymmetric functions whose symmetrizations are Molev’s dual Schur functions [12]. Molev described some properties of dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions and a multiplication rule for the dual Schur functions. Schur functions are an old subject and much is known about them. They are studied in relation to many different subjects from a number of different points of view. We follow the Lascoux–Sch¨ utzenberger approach, viewing Sch¨ ur functions as (symmetric) special cases of Schubert polynomials. From this point of view, it is natural to ask how one can define a larger set of nonsymmetric functions, which will include Molev’s dual Schur functions as their symmetric counterparts. This theme is the main focus of our work. On the algebraic geometry side, we obtain a duality formula for the Schubert classes in Grassmannians in terms of rational Schubert (key) polynomials (Proposition 16). We would also like to point out that a dominant rational Schubert polynomial can be described as a configuration of lines as in [5]: in this work, Fomin and Krillov .
