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,,x nx_1,\ldots,x_n with coefficients which are rational functions of two additional variables qq and tt.

Given a partition λ\lambda, one defines a shift operator T q,x iT_{q,x_i} which maps f=f(x 1,,x n)f = f(x_1,\ldots, x_n) to f(x 1,,x i1,qx i,x i+1,,x n)f(x_1,\ldots, x_{i-1}, q x_i, x_{i+1},\ldots,x_n) and the operators D rD_r, r=0,1,,nr = 0, 1, \ldots, n via

D r=t r(r1)2 I{1,,n},|I|=r iI,jItx ix jx ix j iIT q,x i, 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:= r=0 nD ru rD := \sum_{r=0}^n D_r u^r.

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

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

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

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