Journal article
k-Schur expansions of Catalan functions
Advances in mathematics (New York. 1965), v 371, 107209
16 Sep 2020
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].
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Details
- Title
- k-Schur expansions of Catalan functions
- Creators
- Jonah Blasiak - Drexel UniversityJennifer Morse - University of VirginiaAnna Pun - Drexel UniversityDaniel Summers - Drexel University
- Publication Details
- Advances in mathematics (New York. 1965), v 371, 107209
- Publisher
- Elsevier
- Grant note
- DMS-1600391; DMS-1833333 / NSF (https://doi.org/10.13039/100000001)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000549165500001
- Scopus ID
- 2-s2.0-85086785452
- Other Identifier
- 991019169700804721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics