$K$ - $k$ -Schur functions are $k$ -Schur positive

Haojun Bai, Peter L. Guo

Source record

Source: arXiv

Published: Oct 1, 2026

arXiv: 2610.01708

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Source abstract

The $k$ -Schur functions (resp., $K$ - $k$ -Schur functions) are the symmetric function representatives of Schubert classes in the homology (resp., $K$ -homology) of the affine Grassmannian of type $A$ . We prove that every $K$ - $k$ -Schur function is a nonnegative integer linear combination in the basis of $k$ -Schur functions. This settles a conjecture by Lam, Schilling and Shimozono. The proof relies on realizations of $k$ -Schur and $K$ - $k$ -Schur functions as Catalan and Katalan functions, respectively.

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$K$-$k$-Schur functions are $k$-Schur positive — Mathematical Frontier Network