Combinatorial formula for higher-rank non-symmetric Macdonald polynomials

Construct a combinatorial formula for the higher-rank non-symmetric Macdonald polynomials that generalizes the Haglund–Haiman–Loehr formula for ordinary non-symmetric Macdonald polynomials.

Background

The paper constructs higher-rank non-symmetric Macdonald polynomials as weight vectors in higher-rank polynomial representations of double affine Hecke algebras. These polynomials generalize the usual non-symmetric Macdonald polynomials by incorporating several sets of variables and independent parameters.

Although the paper establishes their existence, uniqueness, weight structure, and triangular expansions, it does not provide a direct combinatorial formula. The authors identify a formula generalizing the Haglund–Haiman–Loehr formula as a conjectural direction and mention compression methods of Guo–Ram as a possible tool.