k-Schur expansions of Catalan functions

2018/11/06 by Blasiak, Jonah, Morse, Jennifer, Pun, Anna +1 · 1 citation
#14N15 #14N35 #33D52 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Primary: 05E05 #Quantum Algebra (math.QA) #Secondary: 05A30

paper · doi:10.48550/arxiv.1811.02490

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 of Lapointe, Lascoux, and Morse; (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, proving a substantial case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that k-Schur functions defined in terms of k-split polynomials agree with strong tableau k-Schur functions.

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• Weighted K-k-Schur functions and their application to the K-k-Schur alternating conjecture

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