A Combinatorial Proof That Schubert vs. Schur Coefficients Are Nonnegative

We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf’s theory of dual equivalence to show that a quasisymmetric function of Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants of the Grassmannian. Dedicated to the memory of Alain Lascoux Introduction A Littlewood-Richardson coefficient is the multiplicity of an irreducible representation of the general linear group in a tensor product of two irreducible representations, and is thus a nonnegative integer. Littlewood and Richardson conjectured a formula for these coefficients in 1934 [25], which was proven in the 1970’s by Thomas [33] and Schützenberger [31]. Since Littlewood-Richardson coefficients may be defined combinatorially as the coefficients of Schur functions in the expansion of a product of two Schur functions, these proofs of the Littlewood-Richardson rule furnish combinatorial proofs of the nonnegativity of Schur structure constants. The Littlewood-Richardson coefficients are also the structure constants for expressing products in the cohomology of a Grassmannian in terms of its basis of Schubert classes. Independent of the Littlewood-Richardson rule, these Schubert structure constants are known to be nonnegative integers through geometric arguments. The integral cohomology ring of any flag manifold has a Schubert basis and again geometry implies that the corresponding Schubert structure constants are nonnegative integers. These cohomology rings and their Schubert bases have combinatorial models, and it remains an open problem to give a combinatorial proof that the Schubert structure constants are nonnegative. We give such a combinatorial proof of nonnegativity for a class of Schubert structure constants in the classical flag manifold. These are the constants that occur in the product of an arbitrary Schubert class by one pulled back from a Grassmannian projection. They are defined combinatorially as the product of a Schubert polynomial [22] by a Schur symmetric polynomial; we call them Schubert vs. Schur coefficients. As the Schubert 2010 Mathematics Subject Classification. 05E05, 14M15.