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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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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
163:, 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.
223:, such that any dominant integral weight is a non-negative integer linear combination of the fundamental weights. The corresponding irreducible representations are the
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is sometimes loosely used to refer to a smallest-dimensional faithful representation, though this is also often called the
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representation (a term referring more to the history, rather than having a well-defined mathematical meaning).
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of the Lie group consisting of the dominant integral weights. It can be proved that there exists a set of
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316:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
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Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
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89: − 1 fundamental representations are the wedge products
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273:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
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129:{\displaystyle \operatorname {Alt} ^{k}\ {\mathbb {C} }^{n}}
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34:irreducible finite-dimensional representation
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77:In the case of the special unitary group
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267:Representation theory. A first course
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219:, indexed by the vertices of the
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170:of the simple Lie group of type
155:of the twofold cover of an odd
40:Lie group or Lie algebra whose
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271:Graduate Texts in Mathematics
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200:are indexed by their highest
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225:fundamental representations
188:irreducible representations
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238:fundamental representation
30:fundamental representation
279:10.1007/978-1-4612-0979-9
312:Hall, Brian C. (2015),
74:of the defining module.
168:adjoint representation
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379:Representation theory
148: − 1.
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18:representation theory
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68:general linear group
217:fundamental weights
153:spin representation
138:alternating tensors
66:In the case of the
50:classical Lie group
136:consisting of the
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46:fundamental weight
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288:978-0-387-97495-8
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72:exterior products
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346:Proposition 8.35
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26:Lie algebras
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263:Harris, Joe
182:Explanation
55:Élie Cartan
374:Lie groups
368:Categories
252:References
232:Other uses
161:spin group
159:, the odd
38:semisimple
22:Lie groups
356:Hall 2015
344:Hall 2015
305:246650103
198:Lie group
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332:Specific
265:(1991).
246:defining
242:standard
61:Examples
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211:in the
202:weights
195:compact
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301:OCLC
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