Original Lapointe–Lascoux–Morse refined Macdonald positivity

Prove the refined Macdonald positivity conjecture of Lapointe–Lascoux–Morse for the original k‑Schur functions: for every partition lambda and integer k ≥ 1, show that the transformed Macdonald polynomial \widetilde{H}_\lambda(X;q,t) expands in the Lapointe–Lascoux–Morse k‑Schur basis \{s^{(k)}_\mu(X;t)\}_\mu with coefficients in Z_{\ge 0}[q,t], i.e., \widetilde{H}_\lambda(X;q,t) = \sum_\mu K^{(k)}_{\lambda,\mu}(q,t)\, s^{(k)}_\mu(X;t) with K^{(k)}_{\lambda,\mu}(q,t) \in Z_{\ge 0}[q,t].

Background

The classical Macdonald positivity conjecture (positivity of Schur coefficients of transformed Macdonald polynomials) was proved by Haiman. Lapointe–Lascoux–Morse (2003) introduced k‑Schur functions and formulated a conjectural refinement asserting nonnegative expansions of transformed Macdonald polynomials in the k‑Schur basis they defined.

This article proves a refined positivity statement in the Chen–Haiman framework for k‑Schur functions (via Catalan symmetric functions), but the authors note that the original Lapointe–Lascoux–Morse formulation remains unresolved. The difference stems from distinct definitions of k‑Schur functions in the literature.