Faithful representation

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

In more abstract language, this means that the group homomorphism

${\displaystyle \rho :G\to GL(V)}$

is injective (or one-to-one).

Caveat: While representations of ${\displaystyle G}$ over a field ${\displaystyle K}$ are de facto the same as ${\displaystyle K[G]}$ -modules (with ${\displaystyle K[G]}$ denoting the group algebra of the group ${\displaystyle G}$ ), a faithful representation of ${\displaystyle G}$ is not necessarily a faithful module for the group algebra. In fact each faithful ${\displaystyle K[G]}$ -module is a faithful representation of ${\displaystyle G}$ , but the converse does not hold. Consider for example the natural representation of the symmetric group ${\displaystyle S_{n}}$ in ${\displaystyle n}$ dimensions by permutation matrices, which is certainly faithful. Here the order of the group is ${\displaystyle n}$ ! while the ${\displaystyle n\times n}$ matrices form a vector space of dimension ${\displaystyle n^{2}}$ . As soon as ${\displaystyle n}$ is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since ${\displaystyle 24>16}$ ); this relation means that the module for the group algebra is not faithful.

Properties

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

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

(the ${\displaystyle n}$ -th tensor power of the representation ${\displaystyle V}$ ) for a sufficiently high ${\displaystyle n}$ .

References

Hazewinkel, Michiel, ed. (2001) [1994], "faithful representation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4