Combinatorial aspects of affine Schubert calculus

The k-Schur functions arose in our study of an open problem on Macdonald polynomials. We will see how these functions play the Schur role in a combinatorial sense by refining classical ideas in symmetric function theory such as Pieri rules, Kostka numbers, the Young lattice, and Young tableaux.
We will also find that the k-Schur functions play a geometric role that mimics the Schur function role. While Schur functions describe the cohomology of the Grassmannian, it turns out that k-Schur functions desribe its quantum cohomology. Consequently, we can show their Littlewood-Richardson coefficients are 3-point Gromov-Witten invariants.
Very recently it was proven by Lam that k-Schur functions give the Schubert basis for the homology of the loop (affine) Grassmannian. We will finish with a discussion of our new results which provide the Schubert basis for the cohomology of the loop Grassmannian, and give the affine Pieri rule for multiplying a special Schubert class with an arbitrary one.
Along the way, we will show how k-Schur functions provide an approach to combinatorial and representation theoretic open problems in the theory of Macdonald polynomials and in the study of Gromov-Witten invariants.
Collaborators on various parts of this work include Lam, Lapointe, Lascoux, and Shimozono.