Fundamental representation

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The irreducible representations of a simply-connected compact Lie group are indexed by their highest weight. These dominant weights form the lattice points in an orthant in the weight lattice of the Lie group. In particular, there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights.

The corresponding irreducible representations are the fundamental representations of the Lie group. In particular, 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.

In the case of the special unitary group SU(n), the n-1 fundamental representations are the wedge products <math>Alt^k\ {\mathbb C}^n</math> consisting of alternating tensors, for k=1,2,...,n-1.

Outside of Lie group 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).