Schur positivity of multi-parameter Macdonald polynomials

Prove that the coefficients of the multi-parameter Macdonald polynomials H_mu(q, t_1, ..., t_{r-1}) and their tilde versions \widetilde{H}_mu(q, t_1, ..., t_{r-1}) in the Schur basis are polynomials with nonnegative integer coefficients.

Background

The paper defines noncommutative multi-t analogues of Macdonald polynomials by replacing the powers t{1+leg(u)} in the Haglund–HHL framework with independent parameters t_{1+leg(u)}. Their commutative images are symmetric functions that specialize to the ordinary Macdonald polynomials when t_i=ti. The conjecture asks for Schur positivity with coefficients in the polynomial ring over the nonnegative integers.

The authors note that the claim is known when the partition has at most two rows, because the multi-parameter family then reduces to the usual one-parameter Macdonald functions, and also for column partitions of the form (1n).

References

The coefficients of H_\mu(q,{\bf t}) (or \tilde H_\mu(q,{\bf t})) on the Schur basis are polynomials with nonnegative integer coefficients.

— Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation  (2502.09072 - Novelli et al., 13 Feb 2025) in Conjecture in Section 4.1, “Multi-t analogues of the Macdonald polynomials”

The $m$-symmetric Schur functions $s_\Lambda(x;t)$ that we study here were introduced in for the sake of the following conjecture. Let $J_\Lambda(x;q,t)$ be the integral form of $P_\Lambda(x;q,t)$, and extend to $R_m$ the plethysm $X \mapsto X/(1-t)$ relevant to Macdonald polynomials. The $m$-symmetric Macdonald positivity conjecture then asserts that

J_\Lambda\left[\frac{X}{1-t};q,t \right]= \sum_{\Omega} K_{\Omega \Lambda}(q,t) \, s_\Omega(x;t),

with $K_{\Omega \Lambda}(q,t) \in \mathbb N[q,t]$.

— A new characterization of right keys, and the $m$-symmetric Schur functions at $t=0$  (2608.13276 - Lapointe et al., 13 Aug 2026) in Section 1, Introduction, paragraph containing equation (\ref{Kostkaintro})