Editing Fundamental representation

In [[representation theory]] of [[Lie group]]s and [[Lie algebra]]s, a '''fundamental representation''' is an [[irreducible representation|irreducible finite-dimensional representation]] of a [[semisimple Lie algebra|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 product]]s of the defining module. 
* In the case of the special unitary group [[SU(n)|SU(''n'')]], the ''n''&nbsp;&minus;&nbsp;1 fundamental representations are the wedge products <math>\operatorname{Alt}^k\ {\mathbb C}^n</math> consisting of the [[alternating tensor]]s, for ''k''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;''n''&nbsp;&minus;&nbsp;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 a Lie group|adjoint representation]] of the simple Lie group of type [[E8 (mathematics)|E<sub>8</sub>]] is a fundamental representation.

== Explanation ==

The [[Irreducible representation | irreducible representations]] of a [[simply-connected]] [[compact group| compact]] [[Lie group]] are indexed by their highest [[weight (representation theory)|weights]]. These weights are the lattice points in an orthant ''Q''<sub>+</sub> 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.<ref>{{harvnb|Hall|2015}} Proposition 8.35</ref> 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.<ref>{{harvnb|Hall|2015}} See the proof of Proposition 6.17 in the case of SU(3)</ref>

== 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 ==
*{{Fulton-Harris}}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-0-387-40122-5}}.

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[[Category:Lie groups]]
[[Category:Representation theory]]