Fundamental representation

• 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 \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. It can be proved that there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram, such that any dominant integral weight is a non-negative integer linear combination of the fundamental weights. The corresponding irreducible representations are the fundamental representations of the Lie group. From the expansion of a dominant weight in terms of the fundamental weights one can take a corresponding tensor product of the fundamental representations and extract one copy of the irreducible representation corresponding to that dominant weight. == Other uses ==

Outside of Lie theory, the term fundamental representation is sometimes loosely used to refer to a smallest-dimensional faithful representation, though this is also often called the standard or defining representation (a term referring more to the history, rather than having a well-defined mathematical meaning). == References ==