# nLab Macdonald polynomial

Macdonald polynomials are a generalization of a Schur functions; they unify a theory of Hall-Littlewood and Jack polynomials. They form a family of orthogonal polynomials? which are symmetric functions in $x_1,\ldots,x_n$ with coefficients which are rational functions of two additional variables $q$ and $t$.

Given a partition $\lambda$, one defines a shift operator $T_{q,x_i}$ which maps $f = f(x_1,\ldots, x_n)$ to $f(x_1,\ldots, x_{i-1}, q x_i, x_{i+1},\ldots,x_n)$ and the operators $D_r$, $r = 0, 1, \ldots, n$ via

$D_r = t^{\frac{r(r-1)}{2}} \sum_{I\subset \{1,\ldots,n\}, |I| = r} \prod_{i\in I, j\notin I} \frac{t x_i-x_j}{x_i-x_j}\prod_{i\in I} T_{q, x_i},$

and the corresponding generating series $D := \sum_{r=0}^n D_r u^r$.

The Macdonald polynomial $P_\lambda(x;q,t)$ is an eigenfunction of $D$ with the eigenvalue

$\prod_{i=1}^n (1 + u t^{n-i} q^{\lambda_i})$

In the case $q = t$ we get the Schur function $P_\lambda(x; t,t) = s_\lambda(t)$. Similarly, shifted Macdonald polynomials generalize shifted Schur functions.

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