In the study of mathematical groups, a group representation is a "description" of a group as a concrete group of transformations (or automorphism group) of some mathematical object. More formally, "description" means that there is a homomorphism from the group to some automorphism group. A faithful representation is one in which this homomorphism is injective.
Some people use realization for this notion and reserve the term representation for what below are called linear representations.
Representation theory divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are:
Finite groups: group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry. The special case where the representation is over a field of characteristic p and p divides the order of the group, called modular representation theory, has very different properties (see below).
Compact or locally compact topological groups: many of the results of finite group representation theory are proved by averaging over the group. These proofs can carry over to infinite groups if the average is replaced by an integral, which only works if an acceptable notion of integral can be defined. This can be done for locally compact groups, using Haar measure. The resulting theory is a central part of harmonic analysis. The Pontryagin duality describes the theory for commutative groups, as a generalised Fourier transform.
Lie groups: Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and the representation theory is crucial to the application of group theory in those fields. See Representations of Lie groups and algebras.
Non-compact topological groups: The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The semisimple Lie groups have a deep theory, building on the compact case. The complementary solvable Lie groups cannot in the same way be classified. The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods.
Within a given kind of representation theory, results differ depending on the kind of automorphism group that is targeted. One target is the permutation groups. But the most important targets are groups of matrices over some field, or, more generally, groups of invertible linear transformations of a vector space.
The most important case is the field of complex numbers (that is, the representations are homomorphisms to a group of complex matrices or invertible linear transformations of a complex vector space). If the vector space is finite dimensional, then the representations are said to be finite dimensional as well. (Infinite dimensional representations are quite possible; the vector space could be an infinite dimensional Hilbert space, for example.)
The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers. Representations in the finite field case are called modular. Here the characteristic of the field is quite significant; many theorems depend on the order of the group not dividing the characteristic of the field.
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2 Linear representation |
A set S is said to be a set-theoretic representation of a group G if there is a function, ρ from G to SS, the set of functions from S to S such that
.
Equivalently, a representation is a group homomorphism from G to the permutation group of S.
See Group action.
A linear representation is a special case of a set representation with additional structure.
In abstract algebra, a representation of a finite group G is a group homomorphism from G to the general linear group GL(n,C) of invertible complex n-by-n matrices. The study of such representations is called representation theory.
Representation theory is important because it enables the reduction of some group theory problems to linear algebra, which has a very well-understood theory.
There is an analogue of this theory for many important kinds of infinite groups; see Representations of Lie groups and algebras and Peter-Weyl theorem for compact topological groups.
A representation by projective transformations (see projective representation) can be described as a linear representation up to scalar matrices. These representations occur naturally, also.
We could also have affine representations. For example, the Euclidean group acts affinely upon Euclidean space.
Consider the complex number u = exp(2πi/3) which has the property u3 = 1. The cyclic group C3 = {1, u, u2} has a representation ρ given by:
This representation is said to be faithful, because ρ is a one-to-one map.
Two representations ρ1 and ρ2 are said to be equivalent if the
matrices only differ by a change of basis, i.e. if there exists A in GL(n,C) such that for all x in G: ρ1(x) = Aρ2(x)A-1. For example, the representation of C3
given by the matrices:
Every square n-by-n matrix describes a linear map from an n-dimensional vector space V to itself (once a basis for V has been chosen). Therefore, every representation ρ: G -> GLn defines a group action on V given by
g.v = (ρ(g))(v) (for g in G, v in V). One may in fact define a representation of a group as an action of that group on some vector space, thereby avoiding the need to choose a basis and the restriction to finite-dimensional vector spaces.
If V has a non-trivial proper subspace W such that W is contained in V, then
the representation is said to be reducible. A reducible representation can be expressed as a
direct sum of subrepresentations (Maschke's theorem) (only for finite groups are reducible representations necessarily decomposable!).
If V has no such subspaces, it is said to be an irreducible representation.
In the example above, the representation given is reducible into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}).
The character of a representation ρ : G -> GLn is the function χ : G -> C which sends g in G to the trace (the sum of the diagonal elements) of the
matrix ρ(g). For example, the character of the representation given above is given by:
χ(1) = 2, χ(u) = 1 + u, χ(u2) = 1 + u2.
If g and h are members of G in the same conjugacy class, then χ(g) = χ(h) for any character; the values of a character therefore have to be specified only for the different conjugacy classes of G. Moreover, equivalent representations have the same characters.
If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the subrepresentations' characters.
The characters of all the irreducible representations of a finite group form a character table, with conjugacy classes of elements as the columns, and characters as the rows. Here is
the character table of C3:
Certain properties of the group G can be deduced from its character table:
Set theoretic representation
Linear representation
Example
(the three matrices are ρ(1), ρ(u) and ρ(u2) respectively).Equivalence of representations
is an equivalent representation to the one shown above.Group actions
Reducibility
Character theory
(1) (u) (u2)
1 1 1 1
χ1 1 u u2
χ2 1 u2 u
The character table is always square, and the rows and columns are orthogonal with respect to the standard inner product on Cm, which allows one to compute character tables more easily. The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1-by-1 matrices containing the entry 1).
The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the
dihedral group of 8 elements (D8) have the same character table.