In
mathematics, especially in an area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
known as
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a faithful representation ρ of a
group on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
is a
linear representation in which different elements of are represented by distinct
linear mappings .
In more abstract language, this means that the
group homomorphism
:
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
(or
one-to-one
One-to-one or one to one may refer to:
Mathematics and communication
*One-to-one function, also called an injective function
*One-to-one correspondence, also called a bijective function
*One-to-one (communication), the act of an individual comm ...
).
''Caveat:'' While representations of over a
field are ''de facto'' the same as -
modules (with denoting the
group algebra of the group ), a faithful representation of is not necessarily a
faithful module for the group algebra. In fact each faithful -module is a faithful representation of , but the
converse does not hold. Consider for example the natural representation of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
in dimensions by
permutation matrices, which is certainly faithful. Here the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of the group is while the
matrices form a vector space of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
. As soon as is at least 4, dimension counting means that some
linear dependence must occur between permutation matrices (since ); this relation means that the module for the group algebra is not faithful.
Properties
A representation of a
finite group over an
algebraically closed field of
characteristic zero is faithful (as a representation)
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
every
irreducible representation of occurs as a
subrepresentation of (the -th symmetric power of the representation ) for a sufficiently high . Also, is faithful (as a representation) if and only if every irreducible representation of occurs as a subrepresentation of
:
(the -th tensor power of the representation ) for a sufficiently high .
[W. Burnside. Theory of groups of finite order. Dover Publications, Inc., New York, 1955. 2d ed. (Theorem IV of Chapter XV)]
References
Representation theory
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