I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the symmetric group.

To make this post more educational, I will define these polynomials a bit. Consider the 2-parameter family of Macdonald operators (indexed by powers of the indeterminate $X$) for root system $A_n$, on a symmetric polynomial $f$ with $x = (x_1, \ldots, x_n)$:

$D(X;t,q) = a_\delta(x)^{-1} \sum_{\sigma \in S_n} \epsilon(\sigma) x^{\sigma \delta}\prod_{i=1}^n (1 + X t^{(\sigma \delta)_i} T_i),$

(mathoverflow doesn't seem to parse $T_{q,x_i}$ in the formula above, so I had to use the shorter symbol $T_i$, which depends on q).

where $\delta$ is the partition $(n-1,n-2,\ldots, 1,0)$, $a_\delta(x) = \prod_{1 \le i < j \le n} (x_i - x_j)$ is the Vandermonde determinant (in general $a_\lambda(x)$ is the determinant of the matrix $(a_i^{\lambda_j})_{i,j \in [n]}$).

$x^{\sigma \delta}$ means $x_1^{(\sigma \delta)_1} x_2^{(\sigma \delta)_2} \ldots x_n^{(\sigma \delta)_n}$.

Also $(\sigma \delta)_i$ denotes the $\sigma(i)$-th component of $\delta$, namely $n-i$.

Finally the translation operator $T_i = T_{q,x_i}$ is defined as $$ T_{q,x_i}f(x_1, \ldots, x_n) = f(x_1, \ldots, x_{i-1}, q x_i , x_{i+1} ,\ldots, x_n).$$

I like to think of the translation operator as the quantized version of the differential operator $I + \partial_i$, where $q$ is analogous to the Planck constant(?).

If we write $D(X;q,t) = \sum_{r=0}^n D_{n-r}(q,t) X^r$, then Macdonald polynomials $p_\lambda(q,t)$ are simply simultaneous eigenfunctions of these operators. When $q=t$ they become Schur polynomials, defined by $s_\lambda = a_{\delta +\lambda} / a_\delta$. When $q= t^\alpha$ and $t \to 1$, we get Jack symmetric polymomials, which are eigenfunctions of a Metropolis random walk on the set of all partitions that converge to the so-called Ewens sampling measure, which assigns probability proportional $\alpha^{\ell(\lambda)} z_\lambda^{-1}$. When $q = 0$, they become the Hall-Littlewood polynomials and when $t=1$ they become the monomial symmetric polynomials etc.

I was told repeatedly by experts that Macdonald polynomials exhaust all previous symmetric polynomial bases in some sense. Does anyone know a theorem that says that every family of symmetric polynomial under some conditions can be obtained from Macdonald polynomials by specializing the $q$ and $t$?