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Irreducible representation
Type of group and algebra representation / From Wikipedia, the free encyclopedia
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In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure
is a nonzero representation that has no proper nontrivial subrepresentation
, with
closed under the action of
.
Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible.