For every positive integer k and every k-bounded partition lambda, the K-k-Schur function mathsfKSchur^(k)_lambda is a nonnegative integer linear combination of k-Schur functions; equivalently, there exist coefficients c^(k)_lambda mu∈Z_≥ 0, indexed by k-bounded partitions mu, such that mathsfKSchur^(k)_lambda=∑_muc^(k)_lambda mu,mathsfSchur^(k)_mu.

Lam–Schilling–Shimozono K-k-Schur positivity conjecture

Lam, Schilling, and Shimozono introduced the relevant $K$ -theoretic $k$ -Schur functions in 2009 and conjectured positivity properties for them. The conjecture concerns positivity in the $K$ -homology Schubert calculus of the affine Grassmannian.

Known results

Lam, Schilling, and Shimozono (2009): constructed the $K$ -homology Schubert basis and identified it, for $G=\mathrm{SL}_n$ , with $K$ - $k$ -Schur functions, while leaving the relevant positivity conjectured.
Blasiak, Morse, and Seelinger (2022; revised 2023): proved a related realization of closed $k$ -Schur Katalan functions as affine-Grassmannian $K$ -homology Schubert representatives.

October 2026 claimed proof

Haojun Bai and Peter L. Guo claim that $K$ - $k$ -Schur functions are $k$ -Schur positive, using Catalan and Katalan-function realizations. This directly addresses the Lam–Schilling–Shimozono positivity conjecture, but the preprint is unrefereed and no independent assessment was found.

Current status (as of October 2026): The conjecture has a direct unrefereed proof claim by Bai and Guo, but its correctness remains unverified.

Lam–Schilling–Shimozono K-k-Schur positivity conjecture

Primary source

Additional references

Progress summary

Solutions 0