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 Gra...

We give a combinatorial rule for calculating the coe cients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coe cients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

In this paper, we consider the Schur complement of a subtransformation of a linear transformation defined on the product of two finite dimensional real Hilbert spaces, and in particular, on two Euclidean Jordan algebras. We study complementarity properties of linear transformations that are inherited by principal subtransformations, principal pivot transformations, and Schur complements.

In a recent article [8], Gowda and Sznajder studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any S...