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arXiv · 2007.04952

Demazure crystals and the Schur positivity of Catalan functions

Jonah Blasiak·Jennifer Morse·Anna Pun
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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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Jonah Blasiak, Jennifer Morse, Anna Pun. 2020-07-09. Demazure crystals and the Schur positivity of Catalan functions. https://arxiv.org/abs/2007.04952

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