A proof of the $m$-Symmetric Macdonald positivity at $t=1$

Abstract

We prove, in the case $t=1$, the extension to the $m$-symmetric world of the original Macdonald positivity conjecture. This is achieved by giving a combinatorial interpretation of the Kostka coefficients $K_{ΩΛ}(q,1)$ in terms of standard fillings of the diagram associated to the $m$-partition $Ω$. This interpretation generalizes the one in the usual Macdonald case, which is given by a major index statistic on standard tableaux.