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

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