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Catalan functions and \(k\)-Schur positivity. (English) Zbl 1423.05192
Summary: 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.

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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