The objects in a number of mathematical categories (groups, C*-algebras, σ-fields of sets are but three examples) can be represented in a variety of ways by means of operators on Hilbert spaces. For example, the permutation group on n letters acts in an obvious way to permute the elements of an orthonormal basis for an n-dimensional Hilbert space, and this gives rise to a representation of that group as a group of unitary operators. Multiplicity theory has to do with the classification of these representations. This classification problem amounts to finding, for a given object, a suitable collection of representation invariants which will allow one to determine when any two of its representations are, or are not, geometrically the “same”—that is, unitarily equivalent. It goes without saying that in order to classify representations one first has to find them (all of them), so an adequate solution to this vague problem should contain a procedure for constructing all possible representations of the object (more precisely, at least one representative from every unitary equivalence class) in usable and concrete terms.
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