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Preprint
Demazure crystals and the Schur positivity of Catalan functions
arXiv (Cornell University)
09 Jul 2020
Abstract
Catalan functions, the graded Euler characteristics of certain vector bundles
on the flag variety, are a rich class of symmetric functions which include
$k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that
Catalan functions indexed by partition weight are the characters of
$U_q(\widehat{\mathfrak{sl}}_\ell)$-generalized Demazure crystals as studied by
Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for
these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our
approach more generally gives key positive formulas for graded Euler
characteristics of certain vector bundles on Schubert varieties by matching
them to characters of generalized Demazure crystals.
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Details
- Title
- Demazure crystals and the Schur positivity of Catalan functions
- Creators
- Jonah BlasiakJennifer MorseAnna Pun
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862257304721