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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 $x_1,\dots ,x_n$ 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 $f=f(x_1,\dots ,x_n)$ to $f(x_1,\dots ,x_{i-1},q{x}_i,x_{i+1},\dots ,x_n)$ and the operators $D_r$, $r=0,1,\dots ,n$ via

$$D_i=t^{\frac{r(r-1)}{2}}\sum _{I\subset \{1,\dots ,n\},\mid I\mid =r}\prod _{i\in I,j\notin I}\frac{t{x}_i-x_j}{x_i-x_j}\prod _{i\in I}{T}_{q,x_i},$$D_i = 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

$$\prod _{i=1}^n(1+u{t}^{n-i}{q}^{{\lambda }_i})$$\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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category: algebra, combinatorics