𝑚-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures

We study the space, R m Rₘ, of m m -symmetric functions consisting of polynomials that are symmetric in the variables x m + 1, x m + 2, x m + 3, … x₌+₁, x₌+₂, x₌+₃, but have no special symmetry in the variables x 1, …, x m x₁, , xₘ. We obtain m m -symmetric Macdonald polynomials by t t -symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of R m Rₘ. We define m m -symmetric Schur functions through a somewhat complicated process involving their dual basis, multi-Schur functions, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the m m -symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of m m -symmetric Schur functions. We obtain relations on the (q, t) (q, t) -Koska coefficients K Ω Λ (q, t) K (q, t) in the m m -symmetric world, and show in particular that the usual (q, t) (q, t) -Koska coefficients are special cases of the K Ω Λ (q, t) K (q, t) ’s. Finally, we show that when m m is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials.