- Fundamental representation
In

representation theory ofLie group s andLie algebra s, a**fundamental representation**is an irreducible finite-dimensional representation of a semisimple Lie groupor Lie algebra whosehighest weight is afundamental weight . For example, the defining module of aclassical 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 toÉlie 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 areexterior power s of the defining module.

* In the case of the special unitary groupSU(n) , the n-1 fundamental representations are the wedge products $Alt^k\; \{mathbb\; C\}^n$ consisting of thealternating tensor s, for k=1,2,...,n-1.

* Thespin representation of the twofold cover of an oddorthogonal group , the oddspin 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.

* Theadjoint representation of the simple Lie group of type E_{8}is a fundamental representation.**Explanation**The irreducible representations of a

simply-connected compactLie group are indexed by their highest weights. These weights are the lattice points in an orthant "Q"_{+}in theweight lattice of the Lie group consisting of the dominant integral weights. It can be provedthat there exists a set of "fundamental weights", indexed by the vertices of theDynkin diagram , such that any dominant integral weight is a non-negative integer linear combinations 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**

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