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A Markov chain on the symmetric group and Jack symmetric functions

✍ Scribed by Phil Hanlon

Publisher: Elsevier Science
Year: 1992
Tongue: English
Weight: 872 KB
Volume: 99
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(92)90370-u

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✦ Synopsis

Hanlon, P., A Markov chain on the symmetric group and Jack symmetric functions, Discrete Mathematics 99 (1992) 123-140. Diaconis and Shahshahani studied a Markov chain Wf(l) whose states are the elements of the symmetric group S,. In W,(l), you move from a permutation n to any permutation of the form a(i, j) with equal probability. In this paper we study a deformation W,(a) of this Markov chain which is obtained by applying the Metropolis algorithm to Wf(l). The stable distribution of W,(a) is 6-C(Z) where C(A) denotes the number of cycles of x. Our main result is that the eigenvectors of the transition matrix of W,(a) are the Jack symmetric functions.

We use facts about the Jack symmetric functions due to Macdonald and Stanley to obtain precise estimates for the rate of convergence of W,( (u) to its stable distribution.

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