# 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

[edit]- 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 E
_{8}is a fundamental representation.

## Explanation

[edit]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.^{[1]} 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.^{[2]}

## Other uses

[edit]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

[edit]- Fulton, William; Harris, Joe (1991).
*Representation theory. A first course*. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. - Hall, Brian C. (2015),
*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-0-387-40122-5.

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