Yang–Baxter trace realization of Macdonald polynomials

Establish that, for every partition mu, the equivariant trace of the Yang–Baxter element Y_{sigma_mu}(v_mu), normalized by the coefficient of s_n, equals the Macdonald polynomial \widetilde{H}_mu((1-q)X;q;t).

Background

For each partition mu, the paper constructs a permutation sigma_mu and a vector v_mu of spectral parameters from the Ferrers diagram of mu. It then considers the corresponding Yang–Baxter element in the Hecke algebra and its equivariant trace.

The conjecture proposes a uniform representation-theoretic realization of the ordinary Macdonald polynomials through these traces. The authors report verification for all partitions of size at most 8, but do not prove the general statement.

References

The equivariant trace of the Yang-Baxter element $Y_{\sigma_\mu}(v_\mu)$ normalized by the coefficient of $s_n$ is equal to the Macdonald polynomial $\tilde H_\mu((1-q)X;q;t)$.

— 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”