The
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 ...
of
groups
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 ide ...
is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on
operations of groups on
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 can ...
s. Nevertheless, groups acting on other groups or on
sets are also considered. For more details, please refer to the section on
permutation representations.
Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over
fields of
characteristic zero. Because the theory of
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 .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over
Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
to examine the structure of groups. There are also applications in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
and
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. For example, representation theory is used in the modern approach to gain new results about automorphic forms.
Definition
Linear representations
Let
be a
–vector space and
a finite group. A linear representation of
is a
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)
w ...
Here
is notation for a
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, and
for an
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
. This means that a linear representation is a map
which satisfies
for all
The vector space
is called representation space of
Often the term representation of
is also used for the representation space
The representation of a group in a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
instead of a vector space is also called a linear representation.
We write
for the representation
of
Sometimes we use the notation
if it is clear to which representation the space
belongs.
In this article we will restrict ourselves to the study of finite-dimensional representation spaces, except for the last chapter. As in most cases only a finite number of vectors in
is of interest, it is sufficient to study the
subrepresentation In 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 algeb ...
generated by these vectors. The representation space of this subrepresentation is then finite-dimensional.
The degree of a representation is the
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 coord ...
of its representation space
The notation
is sometimes used to denote the degree of a representation
Examples
The trivial representation is given by
for all
A representation of degree
of a group
is a homomorphism into the multiplicative
group As every element of
is of finite order, the values of
are
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
. For example, let
be a nontrivial linear representation. Since
is a group homomorphism, it has to satisfy
Because
generates
is determined by its value on
And as
is nontrivial,
Thus, we achieve the result that the image of
under
has to be a nontrivial subgroup of the group which consists of the fourth roots of unity. In other words,
has to be one of the following three maps:
:
Let
and let
be the group homomorphism defined by:
:
In this case
is a linear representation of
of degree
Permutation representation
Let
be a finite set and let
be a group acting on
Denote by
the group of all permutations on
with the composition as group multiplication.
A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations - where groups act on vector spaces instead of on arbitrary finite sets - we have to proceed in a different way. In order to construct the permutation representation, we need a vector space
with
A basis of
can be indexed by the elements of
The permutation representation is the group homomorphism
given by
for all
All linear maps
are uniquely defined by this property.
Example. Let
and
Then
acts on
via
The associated linear representation is
with
for
Left- and right-regular representation
Let
be a group and
be a vector space of dimension
with a basis
indexed by the elements of
The left-regular representation is a special case of the
permutation representation
In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...
by choosing
This means
for all
Thus, the family
of images of
are a basis of
The degree of the left-regular representation is equal to the order of the group.
The right-regular representation is defined on the same vector space with a similar homomorphism:
In the same way as before
is a basis of
Just as in the case of the left-regular representation, the degree of the right-regular representation is equal to the order of
Both representations are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
via
For this reason they are not always set apart, and often referred to as "the" regular representation.
A closer look provides the following result: A given linear representation
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the left-regular representation if and only if there exists a
such that
is a basis of
Example. Let
and
with the basis
Then the left-regular representation
is defined by
for
The right-regular representation is defined analogously by
for
Representations, modules and the convolution algebra
Let
be a finite group, let
be a commutative
ring and let