Catalan functions and 𝑘-Schur positivity

Blasiak, Jonah; Morse, Jennifer; Pun, Anna; Summers, Daniel

doi:10.1090/jams/921

Catalan functions and $k$-Schur positivity
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by Jonah Blasiak, Jennifer Morse, Anna Pun and Daniel Summers

J. Amer. Math. Soc. 32 (2019), 921-963

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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