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| In [[mathematics]], especially in the area of [[abstract algebra]] known as [[representation theory]], a '''faithful representation''' ρ of a [[group (mathematics)|group]] ''G'' on a [[vector space]] ''V'' is a [[linear representation]] in which different elements ''g'' of ''G'' are represented by distinct linear mappings ρ(''g'').
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| In more abstract language, this means that the [[group homomorphism]] | |
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| :ρ: ''G'' → ''GL''(''V'')
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| is [[injective]]. | |
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| ''Caveat:'' While representations of ''G'' over a field ''K'' are ''de facto'' the same as <math>K[G]</math>-modules (with <math>K[G]</math> denoting the [[Group_ring#Group_algebra_over_a_finite_group|group algebra]] of the group ''G''), a faithful representation of ''G'' is not necessarily a [[faithful module]] for the group algebra. In fact each faithful <math>K[G]</math>-module is a faithful representation of ''G'', but the converse does not hold. Consider for example the natural representation of the [[symmetric group]] ''S''<sub>''n''</sub> in ''n'' dimensions by [[permutation matrices]], which is certainly faithful. Here the order of the group is ''n''! while the ''n''×''n'' matrices form a vector space of dimension ''n''<sup>2</sup>. As soon as ''n'' is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16); this relation means that the module for the group algebra is not faithful.
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| ==Properties==
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| A representation ''V'' of a finite group ''G'' over an algebraically closed field ''K'' of characteristic zero is faithful (as a representation) if and only if every irreducible representation of ''G'' occurs as a subrepresentation of ''S''<sup>''n''</sup>''V'' (the ''n''-th symmetric power of the representation ''V'') for a sufficiently high ''n''. Also, ''V'' is faithful (as a representation) if and only if every irreducible representation of ''G'' occurs as a subrepresentation of
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| : <math>V^{\otimes n}=\underbrace{V\otimes V\otimes \cdots\otimes V}_{n\text{ times}}</math>
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| (the ''n''-th tensor power of the representation ''V'') for a sufficiently high ''n''.
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| ==References==
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| {{Springer|id=F/f038170|title=faithful representation}}
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| [[Category:Representation theory]]
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| {{algebra-stub}}
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Latest revision as of 02:59, 22 December 2014
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