Faithful representation

In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group $G$ on a vector space $V$ is a linear representation in which different elements $g$ of $G$ are represented by distinct linear mappings $ρ (g)$ . In more abstract language, this means that the group homomorphism [math]\displaystyle{ \rho: G\to GL(V) }[/math] is injective (or one-to-one).

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 $S n$ in $n$ dimensions by permutation matrices, which is certainly faithful. Here the order of the group is $n!$ while the $n \times n$ matrices form a vector space of dimension $n 2$ . 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 $S n V$ (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

[math]\displaystyle{ V^{\otimes n} = \underbrace{V \otimes V \otimes \cdots \otimes V}_{n\text{ times}} }[/math]

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

References

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

Hazewinkel, Michiel, ed. (2001), "faithful representation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=F/f038170

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

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