K-Schur expansions of Catalan functions

Abstract

We make a broad conjecture about the k-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the k-Schur expansion of (1) Hall-Littlewood polynomials, proving the q=0 case of the strengthened Macdonald positivity conjecture from [24]; (2) the product of a Schur function and a k-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) k-split polynomials, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials [37]. In addition, we prove the conjecture that the k-Schur functions defined via k-split polynomials [25] agree with those defined in terms of strong tableaux [21].