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 subrepresentation closed under the action of . Every finite-dimensional unitary representation on a Hermitian vector space is the direct sum of irreducible representations. As irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), these terms are often confused; however, in general there are many reducible but indecomposable representations, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices.

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