Representation of a compact group(2)
Let be a compact group, let be a Banach space and let be a representation. If is a Hilbert space and is a unitary operator for every , then is called a unitary representation. There always is an equivalent norm in for which is unitary.
Every irreducible unitary representation (cf. Irreducible representation) of a compact group is finite-dimensional. Let be the family of all possible pairwise inequivalent irreducible unitary representations of the group . Every unitary representation of is an orthogonal sum of unique representations , , such that is an orthogonal sum, possibly zero, of a set of representations equivalent to .
If is finite, then the family is also finite and contains as many elements as there are distinct conjugacy classes in (moreover, ). The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory (cf. Finite group, representation of a).
If is a connected, simply-connected, compact Lie group and is its complexification (cf. Complexification of a Lie group), then the description of the family for amounts (by restricting the representations to ) to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group . The latter family, in turn, allows of a complete description by considering highest weights (cf. Representation with a highest weight vector).
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Representation of a compact group(2). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Representation_of_a_compact_group(2)&oldid=21995