Let be the wreath product of a compact group T with the infinite symmetric group . We study the characters of factor representations of finite type of G, and give a formula which expresses all the characters explicitly.
For , any totally ramified cyclic extension of degree of local fields which are finite extensions of the field of -adic numbers, we describe the -module structure of each fractional ideal of explicitly in terms of the indecomposable -modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.
This is an introductory paper about our recent merge of a noncommutative de Finetti type result with representations of the infinite braid and symmetric group which allows us to derive factorization properties from symmetries. We explain some of the main ideas of this approach and work out a constructive procedure to use in applications. Finally we illustrate the method by applying it to the theory of group characters.