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| In [[representation theory]] of [[Lie group]]s and [[Lie algebra]]s, a '''fundamental representation''' is an irreducible finite-dimensional representation of a [[semisimple Lie algebra|semisimple]] Lie group
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| 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.
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| == Examples ==
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| * In the case of the [[general linear group]], all fundamental representations are [[exterior power]]s of the defining module.
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| * In the case of the special unitary group [[SU(n)|SU(''n'')]], the ''n'' − 1 fundamental representations are the wedge products <math>\operatorname{Alt}^k\ {\mathbb C}^n</math> consisting of the [[alternating tensor]]s, for ''k'' = 1, 2, ..., ''n'' − 1.
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| * 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.
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| * 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.
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| == Explanation ==
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| The [[Representation of a Lie group| 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
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| 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 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.
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| == Other uses == | |
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| 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).
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| == References ==
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| {{Fulton-Harris}}
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| [[Category:Lie groups]] | |
| [[Category:Representation theory]] | |
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