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K-k-Schur functions are k-Schur positive

  • Haojun Bai
  • Peter L. Guo
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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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Cite

BibTeXRIS
BibTeX
@misc{bai2026kkschur,
  title = {{$K$-$k$-Schur functions are $k$-Schur positive}},
  author = {Haojun Bai and Peter L. Guo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01708},
  eprint = {2610.01708},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01708},
}
RIS
TY  - UNPB
TI  - K-k-Schur functions are k-Schur positive
AU  - Haojun Bai
AU  - Peter L. Guo
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01708
UR  - https://arxiv.org/abs/2610.01708
AB  - 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.
ER  -