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Group representation
Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which has a very well-understood theory.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "description" means a homomorphism from the group to the automorphism group of the object. If the object is a vector space we have a linear representation. Some people use realization for this notion and reserve the term representation for linear representations. The bulk of this article desribes linear representation theory; see the last section for generalizations.
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Branches of representation theory
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. If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory; this special case has very different properties (see below). See Representation of a finite group.
- 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 be carried over to infinite groups by replacement of the average with an integral, provided that 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. See also: Peter-Weyl theorem.
- 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 their representation theory is crucial to the application of group theory in those fields. See Representations of Lie groups and Representations of Lie 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.
Representation theory also depends heavily on the type of vector space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a Hilbert space, Banach space, etc.).
One must also consider the type of field over which the vector space is defined. The most important case is the field of complex numbers. The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers. In general, algebraically closed fields are easier to handle than non-algebraically closed ones. The characteristic of the field is also quite significant; many theorems depend on the order of the group not dividing the characteristic of the field. Representations over a finite field are called modular representations.
Basic definitions
A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V. It is common practice to refer to V itself as the representation when the homomorphism is clear from context (and, often, even when it isn't).
In the case where V is of finite dimension n it is common to choose a basis for V and identify GL(V) with GL(n, K) the group of n-by-n invertible matrices.
A faithful representation is one in which the homomorphism G → GL(V) is injective.
Simple example
Consider the complex number u = exp(2πi/3) which has the property u^{3} = 1. The cyclic group C_{3} = {1, u, u^{2}} has a representation ρ on C^{2} given by:
(the three matrices are ρ(1), ρ(u) and ρ(u^{2}) respectively). This representation is faithful because ρ is a one-to-one map.
Equivalence of representations
Two representations ρ_{1} and ρ_{2} 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: ρ_{1}(x) = Aρ_{2}(x)A^{-1}. For example, the representation of C_{3} given by the matrices:
is an equivalent representation to the one shown above.
Reducibility
A subspace W of V that is fixed under the group action is called a subrepresentation. If V has a non-zero proper subrepresentation, the representation is said to be reducible. Otherwise, it is said to be irreducible.
Representations of finite groups can always be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem.)
In the example above, the representation given is decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}).
Character theory
The character of a representation ρ : G → GL_{n}C 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(u^{2}) = 1 + u^{2}.
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 C_{3}:
(1) | (u) | (u^{2}) | |
1 | 1 | 1 | 1 |
χ_{1} | 1 | u | u^{2} |
χ_{2} | 1 | u^{2} | u |
The character table is always square, and the rows and columns are orthogonal with respect to inner products on C^{m} (see orthogonality relation ), 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 1×1 matrices containing the entry 1).
Certain properties of the group G can be deduced from its character table:
- The order of G is given by the sum of (χ(1))^{2} over the characters in the table.
- G is abelian if and only if χ(1) = 1 for all characters in the table.
- G has a non-trivial normal subgroup (i.e. G is not a simple group) if and only if χ(1) = χ(g) for some non-trivial character χ in the table and some non-identity element g in G.
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 (D_{8}) have the same character table.
Generalizations
Set-theoretical representations
A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X is given by a function ρ from G to X^{X}, the set of functions from X to X, such that for all g_{1}, g_{2} in G and all x in X:
- ρ(1)[x] = x
- ρ(g_{1}g_{2})[x] = ρ(g_{1})[ρ(g_{2})[x]]
This condition and the axioms for a group imply that ρ(g) is a bijection (or permutation) for all g in G. Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group S_{X} of X.
For more information on this topic see the article on group action.
Representations in other categories
Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G. Given an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(X), the automorphism group of X.
In the case where C is Vect_{K}, the category of vector spaces over a field K, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of G in the category of sets.
For another example consider the category of topological spaces, Top. Representations in Top are homomorphisms from G to the homeomorphism group of a topological space X.
Two types of representations closely related to linear representations are:
- projective representations: in the category of projective spaces. These can be described as "linear representations up to scalar transformations".
- affine representations: in the category of affine spaces. For example, the Euclidean group acts affinely upon Euclidean space.