Irreducible representation

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In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ${\displaystyle (\rho ,V)}$ or irrep of an algebraic structure ${\displaystyle A}$ is a nonzero representation that has no proper nontrivial subrepresentation ${\displaystyle (\rho |_{W},W)}$, with ${\displaystyle W\subset V}$ closed under the action of ${\displaystyle \{\rho (a):a\in A\}}$.

Every finite-dimensional unitary representation on a Hilbert space ${\displaystyle V}$ 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.