Affine dual equivalence and k-Schur positivity

We introduce an analog of dual equivalence for the affine symmetric group which shares many nice properties with dual equivalence on permutations. In particular, we use this relation to define a D graph structure on starred strong tableaux, defined by Lam, Lapointe, Morse and Shimozono, which preserves an additional statistic called spin. As a corollary, we obtain a combinatorial proof of the Schur positivity of k-Schur functions, introduced by Lapointe, Lascoux and Morse. Moreover, there are strong connections between these D graphs and those constructed for Macdonald polynomials which may shed light on the Lapointe-Lascoux-Morse conjecture that Macdonald polynomials expand positively on the k-Schur basis.

This is joint work with Sara Billey at the University of Washington.