1. k-Schur expansions of Catalan functions.
- Author
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Blasiak, Jonah, Morse, Jennifer, Pun, Anna, and Summers, Daniel
- Subjects
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SCHUR functions , *GROMOV-Witten invariants , *SYMMETRIC functions , *CATALAN numbers , *POLYNOMIALS , *COMBINATORICS , *LOGICAL prediction - 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]. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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