Fundamental representation

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 products of the defining module.

In the case of the special unitary group SU(n), the n − 1 fundamental representations are the wedge products

Alt

k

⁡

C

n

{\displaystyle \operatorname {Alt} ^{k}\ {\mathbb {C} }^{n}}

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.

Explanation

The irreducible representations of a simply-connected compact Lie group are indexed by their highest weights. These weights are the lattice points in an orthant Q+ in the weight lattice of the Lie group consisting of the dominant integral weights.