Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro.
The special Schubert calculus is real.
Pieri-type formulas for maximal isotropic Grassmannians via triple intersections
We give an elementary proof of the Pieri-type formula in the cohomology ring of a Grassmannian of maximal isotropic subspaces of an orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The multiplicities (which are powers of 2) in the Pieri-type formula are seen to arise from the intersection of a collection of quadrics with a linear space.
Pieri's formula for flag manifolds and Schubert polynomials
We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri’s formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain...
Experimentation and conjectures in the real Schubert calculus for flag manifolds.
Gale duality for complete intersections
We show that every complete intersection defined by Laurent polynomials in an algebraic torus is isomorphic to a complete intersection defined by master functions in the complement of a hyperplane arrangement, and vice versa. We call systems defining such isomorphic schemes Gale dual systems because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master...
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