A Littlewood-Richardson rule counts pairs of forest RC graphs whose lift product matches a target forest-code and weight c, and the same rule applies to dual bases via a new Schubert bialgebra.
A combinatorial proof that Schubert vs. Schur coefficients are nonnegative
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abstract
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.
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A Littlewood-Richardson Rule for Forest Polynomials via the Schubert Bialgebra
A Littlewood-Richardson rule counts pairs of forest RC graphs whose lift product matches a target forest-code and weight c, and the same rule applies to dual bases via a new Schubert bialgebra.