Faithful representation

In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group Template:Mvar on a vector space Template:Mvar is a linear representation in which different elements Template:Mvar of Template:Mvar are represented by distinct linear mappings Template:Math. 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 Template:Mvar over a field Template:Mvar are de facto the same as Template:Math-modules (with Template:Math denoting the group algebra of the group Template:Mvar), a faithful representation of Template:Mvar is not necessarily a faithful module for the group algebra. In fact each faithful Template:Math-module is a faithful representation of Template:Mvar, but the converse does not hold. Consider for example the natural representation of the symmetric group Template:Math in Template:Mvar dimensions by permutation matrices, which is certainly faithful. Here the order of the group is Template:Math while the Template:Math matrices form a vector space of dimension Template:Math. As soon as Template:Mvar is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since Template:Math); this relation means that the module for the group algebra is not faithful.

Properties

A representation Template:Mvar of a finite group Template:Mvar over an algebraically closed field Template:Mvar of characteristic zero is faithful (as a representation) if and only if every irreducible representation of Template:Mvar occurs as a subrepresentation of Template:Math (the Template:Mvar-th symmetric power of the representation Template:Mvar) for a sufficiently high Template:Mvar. Also, Template:Mvar is faithful (as a representation) if and only if every irreducible representation of Template:Mvar occurs as a subrepresentation of

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

(the Template:Mvar-th tensor power of the representation Template:Mvar) for a sufficiently high Template:Mvar.

References

• Template:Springer

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