The $q,t$ Kostka polynomials are defined to be the coefficients of the big Schur $s_\lambda[X(1-t)]$ in the expansion of the integral form Macdonald polynomials $J_\mu[X;q,t]$. The integral form Macdonald polynomials are obtained by normalizing the Macdonald P-polynomials $P_\lambda(X;q,t)$ in a certain way. Macdonald correctly conjectured that the $J_\mu(X;q,t)$ will have integral monomial coefficients and they expand positively in the big Schur basis. My question is how did he decide to look at that particular normalization of the Macdonald polynomials and why did he decide to expand it in the big Schur basis? Since we are in the $q,t$ world I would have expected to expand in $s_\lambda\bigg[X\dfrac{1-t}{1-q}\bigg]$ or the plain $s_\lambda$ somehow.

Any intuition to come up with the definition of the $J_\lambda$ or the $q,t$-Kostka polynomials will be helpful.



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