Faithful representation

In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group {{mvar|G}} on a vector space {{mvar|V}} is a linear representation in which different elements {{mvar|g}} of {{mvar|G}} are represented by distinct linear mappings {{math|ρ(g)}}.

In more abstract language, this means that the group homomorphism $\rho: G\to GL(V)$ is injective (or one-to-one).

Caveat

While representations of {{mvar|G}} over a field {{mvar|K}} are de facto the same as {{math|K[G]}}-modules (with {{math|K[G]}} denoting the group algebra of the group {{mvar|G}}), a faithful representation of {{mvar|G}} is not necessarily a faithful module for the group algebra. In fact each faithful {{math|K[G]}}-module is a faithful representation of {{mvar|G}}, but the converse does not hold. Consider for example the natural representation of the symmetric group {{math|S_n}} in {{mvar|n}} dimensions by permutation matrices, which is certainly faithful. Here the order of the group is {{math|n!}} while the {{math|n × n}} matrices form a vector space of dimension {{math|n²}}. As soon as {{mvar|n}} is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since {{math|24 > 16}}); this relation means that the module for the group algebra is not faithful.

Properties

A representation {{mvar|V}} of a finite group {{mvar|G}} over an algebraically closed field {{mvar|K}} of characteristic zero is faithful (as a representation) if and only if every irreducible representation of {{mvar|G}} occurs as a subrepresentation of {{math|SⁿV}} (the {{mvar|n}}-th symmetric power of the representation {{mvar|V}}) for a sufficiently high {{mvar|n}}. Also, {{mvar|V}} is faithful (as a representation) if and only if every irreducible representation of {{mvar|G}} occurs as a subrepresentation of

: $V^{\otimes n} = \underbrace{V \otimes V \otimes \cdots \otimes V}_{n\text{ times}}$

(the {{mvar|n}}-th tensor power of the representation {{mvar|V}}) for a sufficiently high {{mvar|n}}.W. Burnside. Theory of groups of finite order. Dover Publications, Inc., New York, 1955. 2d ed. (Theorem IV of Chapter XV)

References

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Category:Representation theory