Affine dual equivalence and $k$-Schur functions

The $k$-Schur functions were first introduced by Lapointe, Lascoux and Morse in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for $k$-Schur functions was given by Lam, Lapointe, Morse, and Shimozono as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type $A$ and at $t=1$ it is equivalent to the $k$-tableaux characterization of Lapointe and Morse. In this paper, we extend Haiman’s dual equivalence relation on standard Young tableaux to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf’s dual equivalence graphs. These graphs display much of the complexity of working with $k$-Schur functions and the interval structure on $\widetilde{S}_{n}/S_{n}$. We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu de taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we state some open problems on other ways to generalize dual equivalence.

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Published 19 February 2013