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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

V ⊗ n = V ⊗ V ⊗ ⋯ ⊗ V ⏟ n  times {\displaystyle V^{\otimes n}=\underbrace {V\otimes V\otimes \cdots \otimes V} _{n{\text{ times}}}}

(the n-th tensor power of the representation V) for a sufficiently high n.¹

• "faithful representation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

References

1. W. Burnside. Theory of groups of finite order. Dover Publications, Inc., New York, 1955. 2d ed. (Theorem IV of Chapter XV) ↩