$m$-symmetric Macdonald polynomials

Abstract

We study non-symmetric Macdonald polynomials whose variables $x_{m+1},x_{m+2},...$ are symmetrized (using the Hecke symmetrization), which we call $m$-symmetric Macdonald polynomials (the case $m=0$ corresponds to the usual Macdonald polynomials). In the space of $m$-symmetric polynomials, we define $m$-symmetric Schur functions (now depending on the parameter $t$) by certain triangularity conditions. We conjecture that the $m$-symmetric Macdonald polynomials are positive (after a plethystic substitution) when expanded in the basis of $m$-symmetric Schur functions and that the corresponding $m-(q,t)$-Kostka coefficients embed naturally into the $m+1-(q,t$)-Kostka coefficients. When $m=1$, an analog of the nabla operator can be defined, which provides a refinement of the bigraded Frobenius series of the space of diagonal harmonics. When $m$ is larger, how to define such a nabla operator is still an open problem.