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Abstract:
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.
This definition has been shown to correspond to the Schubert
basis for the affine Grassmannian. Using this new definition for k-Schur functions,
we prove the symmetry and Schur positivity of k-Schur functions combinatorially using
the theory of dual equivalence graphs. Central to our proof is our discovery of an analog
of dual equivalence for the affine symmetric group. No background in symmetric functions
will be assumed for this talk.
This is joint work with Sara Billey at the University of Washington.
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