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

We study the space, R m R_m , of m m -symmetric functions consisting of polynomials that are symmetric in the variables x m + 1 , x m + 2 , x m + 3 , … xₘ₊₁,xₘ₊₂,xₘ₊₃, but have no special symmetry in the variables x 1 , … , x m x_1, ,x_m . 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 R_m . 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 ) KΩ Λ(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 ) KΩ Λ(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.