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Catalan functions and $ k$-Schur positivity


Authors: Jonah Blasiak, Jennifer Morse, Anna Pun and Daniel Summers
Journal: J. Amer. Math. Soc. 32 (2019), 921-963
MSC (2010): Primary 05E05, 05E10
DOI: https://doi.org/10.1090/jams/921
Published electronically: August 22, 2019
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Abstract: We prove that graded $ k$-Schur functions are $ G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $ k$-Schur functions and resolve the Schur positivity and $ k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.


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Additional Information

Jonah Blasiak
Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
Email: jblasiak@gmail.com

Jennifer Morse
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: morsej@virginia.edu

Anna Pun
Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
Email: annapunying@gmail.com

Daniel Summers
Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
Email: danielsummers72@gmail.com

DOI: https://doi.org/10.1090/jams/921
Keywords: $k$-Schur functions, Schur positivity, branching rule, spin, strong tableaux, generalized Kostka polynomials
Received by editor(s): April 18, 2018
Received by editor(s) in revised form: January 31, 2019
Published electronically: August 22, 2019
Additional Notes: Authors were supported by NSF Grants DMS-1600391 (the first author) and DMS-1833333 (the second author)
Article copyright: © Copyright 2019 American Mathematical Society