Mixing with staircase multiplicity functions

Abstract

Abstract Every subgroup of the symmetric group defines a natural factor of the Cartesian power of a transformation. We calculate the set of values of the spectral multiplicity function of such factors (under certain conditions on the transformation) in terms of the number of orbits of diagonal actions of these subgroups. An analogous statement also holds on the unitary level for operators that preserve 1. In particular, we prove that for every positive integer n , there exists a transformation which is mixing of all orders and has a staircase multiplicity function of length n ; that is, the essential values of the spectral multiplicity function are {1,2,…, n }.