在数学里，表示理论是以线性变换的群来分析一般抽象群的一种技术。相关的介绍请见群表示，此条目则讨论含有有限个元素的群的表示理论。

此条目中的所有线性变换都是有限维的，且除了有另外提起外，都假定为复数。G的表示是一个群同构 ρ:G → GL(n,C)，由 G 至一般线性群 GL(n,C) 的映射。因此，要选定一个表示，则只要将群内的每个元素配定一个方阵，其中方阵的相乘和群元素间的运算会是一样的。

若矩阵是实数的，则称 ρ 是 G 的一个实表示。换句话说， ρ(G) ⊂ GL(n,R)。

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.