We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at t " 0 in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall-Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.
This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.
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