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 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 Template:Mvar over a field Template:Mvar are de facto the same as K[G]-modules (with K[G] 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 K[G]-module is a faithful representation of Template:Mvar, but the converse does not hold. Consider for example the natural representation of the symmetric group S_n in Template:Mvar 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 Template:Mvar 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 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 SⁿV (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

Template:Asbox