Abstract
The two-parameter Macdonald polynomials are a central object of algebraic
combinatorics and representation theory. We give a Markov chain on partitions
of k with eigenfunctions the coefficients of the Macdonald polynomials when
expanded in the power sum polynomials. The Markov chain has stationary
distribution a new two-parameter family of measures on partitions, the inverse
of the Macdonald weight (rescaled). The uniform distribution on permutations
and the Ewens sampling formula are special cases. The Markov chain is a version
of the auxiliary variables algorithm of statistical physics. Properties of the
Macdonald polynomials allow a sharp analysis of the running time. In natural
cases, a bounded number of steps suffice for arbitrarily large k.
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