  The k-Schur functions were first introduced by Lapointe, Lascoux and Morse
(2003) 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 (2010) 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 by Lam (2008), and, at t = 1, it is
equivalent to the k-tableaux characterization of Lapointe and Morse (2007). In
this paper, we extend Haiman's (1992) 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
affine Symmetric Group modulo the Symmetric Group. We introduce the notions of
flattening and squashing skew starred strong tableaux in analogy with jeu da
taquin slides in order to give a method to find all isomorphism types for
affine dual equivalence graphs of rank 4. Finally, we make connections between
k-Schur functions and both LLT and Macdonald polynomials by comparing the
graphs for these functions.
