Fundamental representation
From Wikipedia, the free encyclopedia
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.
- In the case of the general linear group, all fundamental representations are exterior products 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.
