In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in an area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
known as
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, a faithful representation ρ of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
is a
linear representation
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 ...
in which different elements of are represented by distinct
linear mapping
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vec ...
s .
In more abstract language, this means that the
group homomorphism
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
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 of its codomain; that is, implies (equivalently by contraposition, impl ...
(or
one-to-one).
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
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of .
Over an integral domain, a module that has a nonzero annihilator ...
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 grou ...
in dimensions by
permutation matrices
In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...
, which is certainly faithful. Here the
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
of the group is while the
matrices
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the ...
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 coo ...
. As soon as is at least 4, dimension counting means that some
linear dependence
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...
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
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
every
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
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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