Irreducible representation

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

Every finite-dimensional unitary representation on a Hermitian vector space $V$ 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.

单词	Irrep
释义	Irrep 英语百科 Irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation $(\rho, V)$ or irrep of an algebraic structure $A$ is a nonzero representation that has no proper subrepresentation $(\rho\|_W,W), W \subset V$ closed under the action of $\{ \rho(a) : a\in A \}$ . Every finite-dimensional unitary representation on a Hermitian vector space $V$ 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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Irrep

Irreducible representation