A Markov chain on the symmetric group and Jack symmetric functions

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. 1. A Markov chain A number of mathematical and statistical problems lead to the considerations of random walks where the set of states is a finite or continuous group (see Diaconis [l, Chapter 31, for a thorough and entertaining discussion with references). Usually, in these random walks on groups, the transitional probability r(x, y) of going from y to x depends only on the group element xy-’ and in most cases only on the conjugacy class of xy-‘. Diaconis and Shahshahani [2] analyze a proposed card-shuffling procedure using a random walk on the symmetric group S, where the transitional probability ti(o, n) is (5))’ if UK’ is a transposition and 0 otherwise. In their random walk, you move from a permutation Ed to any permutation of the form n(i, j) with equal probability. We will denote this random walk by IV,(l). Let c(n) denote the number of cycles in the disjoint cycle decomposition of n. *The author is grateful to the Sloan Foundation, Oxford University, the University of Michigan and the National Science Foundation for partial support of this work. 0012-365X/92/$05.00