Montonen–Olive duality: Difference between revisions

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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'').
 
In more abstract language, this means that the [[group homomorphism]]
 
:ρ: ''G'' → ''GL''(''V'')
 
is [[injective]].
 
''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''&times;''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.
 
==Properties==
 
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
: <math>V^{\otimes n}=\underbrace{V\otimes V\otimes \cdots\otimes V}_{n\text{ times}}</math>
(the ''n''-th tensor power of the representation ''V'') for a sufficiently high ''n''.
 
==References==
{{Springer|id=F/f038170|title=faithful representation}}
 
[[Category:Representation theory]]
 
{{algebra-stub}}

Revision as of 13:37, 20 April 2013

In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a 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).

In more abstract language, this means that the group homomorphism

ρ: GGL(V)

is injective.

Caveat: While representations of G over a field K are de facto the same as K[G]-modules (with K[G] denoting the 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 K[G]-module is a faithful representation of G, but the converse does not hold. Consider for example the natural representation of the symmetric group Sn 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 n2. 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.

Properties

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 SnV (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

Vn=VVVn times

(the n-th tensor power of the representation V) for a sufficiently high n.

References

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Template:Algebra-stub