Leonardo C. Mihalcea

Giambelli formulae for the equivariant quantum cohomology of the Grassmannian

We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.

QUANTUM K-THEORY OF GRASSMANNIANS

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersec- tions computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring, and show that its structure constants satisfy S3-symmetry. Our for- mula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.

Chern classes of Schubert cells and varieties

We give explicit formulas for the Chern-Schwartz-MacPherson classes of all Schubert varieties in the Grassmannian of d-planes in a vector space, and conjecture that these classes are effective. We prove this is the case for d ≤ 2.

Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds

We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold G/B. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure-Lusztig type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of G/B. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold G/P. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjectures to the torus equivariant setting.