Rectangular multi-t Macdonald polynomials as Yang–Baxter traces

Prove that, for every rectangular partition mu=(k^r), the equivariant trace of the Yang–Baxter element Y_{sigma_mu}(v_mu) is proportional to the multi-t Macdonald polynomial \omega\widetilde{H}_mu((1-q)X;q,t_1,...,t_{r-1}) with the specified spectral-parameter vector v_mu.

Background

The paper proposes a multi-parameter extension of Macdonald polynomials and investigates whether these functions arise from equivariant traces of Yang–Baxter elements in Hecke algebras. Computational evidence covers many small examples, including rectangular shapes such as (2,2,2).

The rectangular case is singled out because the authors expect the construction to generalize particularly naturally there. The conjecture specifies the spectral parameters in terms of q and the ratios of the multi-t variables.

References

Let $\mu=(kr):=(k,k,\ldots,k)$ ($r$ times) be a rectangular partition. The equivariant trace of the Yang-Baxter element $Y_{\sigma_\mu}(v_\mu)$ is proportional to \begin{equation} \omega\tilde H_\mu((1-q))X;q,t_1,\ldots,t_{r-1}) \end{equation} where the vector of spectral parameters is \begin{equation} v_\mu=\left(1,q,\ldots,q{k-1}, \frac{t_{r-1}}{t_{r-2}},\ldots,q{k-1}\frac{t_{r-1}}{t_{r-2}},\frac{t_{r-1}}{t_{r-3}},\ldots,q{k-1}\frac{t_{r-1}}{t_{r-3}},\ldots,t_{r-1},\ldots,q{k-1}t_{r-1}\right). \end{equation}

— Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation  (2502.09072 - Novelli et al., 13 Feb 2025) in Conjecture in Section 4.3, “Yang-Baxter elements and Macdonald polynomials”