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In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations. == Examples == * In the case of the general linear group, all fundamental representations are exterior powers of the defining module. * In the case of the special unitary group SU(''n''), the ''n'' − 1 fundamental representations are the wedge products consisting of the alternating tensors, for ''k'' = 1, 2, ..., ''n'' − 1. * The spin representation of the twofold cover of an odd orthogonal group, the odd spin group, and the two half-spin representations of the twofold cover of an even orthogonal group, the even spinor group, are fundamental representations that cannot be realized in the space of tensors. * The adjoint representation of the simple Lie group of type E8 is a fundamental representation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「fundamental representation」の詳細全文を読む スポンサード リンク
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