Let X be the affine flag manifold of Lie type A_1^1. Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group D) and the torus stable curves in X. Given a fixed point u in D and a degree d = (d_0,d_1) (a pair of nonnegative integers), the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u using a chain of curves of total degree <= d. In this paper we give a formula for these elements, using combinatorics of the affine root system of type A_1^1. This is the result of an undergraduate research project under my supervision.
We find explicit formulas for the curve neighborhoods of Schubert varieties in the G2 flag manifold. We use this information to study the quantum Chevalley formula, and to obtain quantum Schubert polynomials. We also find an explicit generating set for the ideal of relations of the quantum cohomology ring of the G2 flag manifold. This is the result of an undergraduate research project under my supervision.
We find canonical Schubert polynomials for the equivariant cohomology of the flag manifolds of types B,C,D. These generalize - and are a double version of - the polynomials defined earlier by Billey and Haiman. We prove several combinatorial properties of these polynomials, including a positivity property, a certain symmetry, and we find a formula - in terms of factorial Q-Schur functions - for the Schubert polynomials indexed by the longest permutation in a finite Weyl group of the corresponding type.
The "quantum=classical" phenomenon, discovered by Buch-Kresch and Tamvakis stated that a 3-point, genus 0, Gromov-Witten invariant on a (cominuscule) Grassmannian is equal to an ordinary structure constant on an auxiliary homogeneous space. We generalize this statement for equivariant K-theoretic Gromov-Witten invariants. As an application we find structure theorems (Pieri and Giambelli formula) for quantum K-theory of the Grassmannian.
We effectively compute the Chern-Schwartz-MacPherson class of a Schubert cell - hence that of the Schubert variety - in the Grassmannian. We also show that this class is effective in few instances, and conjecture to hold in general.
We continue the study of the positivity of the Chern-Schwartz-MacPherson (CSM) class, from a combinatorial point of view. We show that the CSM classes of cells, indexed by partitions with 2 or 3 rows, are effective. For general partitions, the conjectural positivity of CSM classes is shown to be equivalent to a generalization to families of the classical theorem of Gessel-Viennot-Lindstrom to compute determinants of binomial coefficients.
We study the equivariant quantum cohomology of the Grassmannian. We prove an EQ Pieri-Chevalley rule and give an algorithm to compute the EQ Littlewood-Richardson coefficients.
We prove that the structure constants for the EQ cohomology algebra of a homogeneous space G/P satisfy a positivity property, generalizing the positivity in equivariant cohomology.
We prove an equivariant quantum Chevalley rule in the equivariant quantum cohomology of a homogeneous space X=G/P, i.e. we give a formula to multiply with Schubert classes correponding to divisors. As in the case when X is a Grassmannian, this implies an effective algorithm to compute the structure constants of equivariant quantum cohomology.
It is known that the Schur functions represent the Schubert classes both in classical and quantum cohomology. The main result of this paper is that both equivariant and equivariant quantum Schubert classes are represented by factorial Schur functions.