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This is an introductory paper about our recent merge of a noncommutative deFinetti type result with representations of the infinite braid and symmetricgroup which allows to derive factorization properties from symmetries. Weexplain some of the main ideas of this approach and work out a constructiveprocedure to use in applications. Finally we illustrate the method by applyingit to the theory of group characters.

Noncommutative Independence in the Infinite Braid and Symmetric Group