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We study a multivariate Markov chain on the symmetric group with remarkableenumerative properties. We conjecture that the stationary distribution of thisMarkov chain can be expressed in terms of positive sums of Schubertpolynomials.  This Markov chain is a multivariate generalization of a Markov chainintroduced by the first author in the study of random affine Weyl groupelements.

A Markov chain on the symmetric group which is Schubert positive?