Rational Schubert polynomials (Q2813412)

The authors introduce a new set of combinatorially defined nonsymmetric functions whose symmetrizations are Molev's dual Schur functions. \textit{A. I. Molev} [Electron. J. Comb. 16, No. 1, Research Paper R13, 44 p. (2009; Zbl 1182.05128)] 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. In this work they follow the Lascoux-Schützenberger approach [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)], viewing Schur 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 their work. On the algebraic geometry side, they obtain a duality formula for the Schubert classes in Grassmannians in terms of rational Schubert (key) polynomials. Also they point out that a dominant rational Schubert polynomial can be described as a configuration of lines as in the work of \textit{S. Fomin} and \textit{A. N. Kirillov} [Discrete Math. 153, No. 1--3, 123--143 (1996; Zbl 0852.05078)].