Demazure crystals and the Schur positivity of Catalan functions

TL;DR

This paper proves that Catalan functions, a broad class of symmetric functions, are characters of generalized Demazure crystals, providing Schur positive formulas and settling existing conjectures in the field.

Contribution

It establishes a link between Catalan functions and Demazure crystal characters, offering new Schur positive formulas and resolving prior conjectures.

Findings

Catalan functions are Demazure crystal characters.

Schur positive formulas for Catalan functions are derived.

Conjectures of Chen-Haiman and Shimozono-Weyman are settled.

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.