Abstract: In this talk I will discuss some joint work with Sami Assaf on the k-Schur functions. Historically, the Schur functions have been an important basis of the ring of symmetric functions. About 10 years ago, Lascoux, Lapointe and Morse introduced a new analog of Schur functions which they call k-Schur functions. There are many interesting connections between the k-Schur functions, Schubert varieties, Gromov-Witten invariants, affine Grassmannians, and tableaux combinatorics. For example, Lam showed that these elements represent Schubert classes in the cohomology ring of the affine Grassmannian when the indeterminate t=1. It is conjectured that the Macdonald polynomials expand positively into the basis of k-Schurs for certain k. In this talk, we will discuss the final step toward proving that the k-Schur functions are Schur postive via a computer proof and Assaf's D-graph machinery. The required computation is a halting problem with a termination condition but no upper bound on the number of steps required to stop. Along the way, I will include several ideas for projects in SAGE that would be useful in this area of research.
