Representation theory of finite groups
Encyclopedia
In mathematics
, representation theory
is a technique for analyzing abstract groups
in terms of groups of linear transformation
s. See the article on group representations for an introduction. This article discusses the representation theory of groups that have a finite number of elements.
unless otherwise stated. A representation of G is a group homomorphism
ρ:G → GL(n,C) from G to the general linear group
GL(n,C). Thus to specify a representation, we just assign a square matrix to each element of the group, in such a way that the matrices behave in the same way as the group elements when multiplied together.
We say that ρ is a real representation
of G if the matrices are real. In other words if ρ(G) ⊂ GL(n,R).
of G on the vector space Cn. Moreover this action completely determines ρ. Hence to specify a representation it is enough to specify how it acts on its representing vector space.
Alternatively, the action of a group G on a complex vector space V
induces a left action of group algebra
C[G] on the vector space V, and vice-versa. Hence representations are equivalent to left C[G]-modules.
The group algebra
C[G] is a |G|-dimensional algebra over the complex numbers, on which G acts. (See Peter–Weyl for the case of compact group
s.)
In fact C[G] is a representation for G×G. More specifically, if g1 and g2 are elements of G and h is an element of C[G] corresponding to the element h of G,
[h]=g1 h g2-1.
C[G] can also be considered as a representation of G in three different ways:
these are all to be 'found' inside the G×G action.
Consider for example the dihedral group
D4 of symmetries of a square. This is generated by the two reflection matrices
Here m is a reflection that maps (x,y) to (− x,y), while n maps (x,y) to (y,x). Multiplying these matrices together creates a set of 8 matrices that form the group. As discussed above, we can either think of the representation in terms of the matrices, or in terms of the action on the two-dimensional vector space (x,y).
This representation is faithful - that is, there is a one-to-one correspondence between the matrices and the elements of the group. It is also irreducible, because there is no subspace of (x,y) that is invariant under the action of the group.
; this example is central to digital signal processing
.
All irreducible representations are 1-dimensional (characters), and correspond to sending a generator of G to a root of unity
, not necessarily primitive (the trivial representation sends a generator to 1, for instance).
A function on G is called the time domain representation of the function, while the corresponding expression in terms of characters is called the frequency domain representation of the function: changing from the time domain description to the frequency domain description is called the discrete Fourier transform, and the opposite direction is called the inverse discrete Fourier transform.
The character table, which in this case is the matrix of the transform, is the
DFT matrix
, which is, up to normalization factor, the Vandermonde matrix for the nth roots of unity; the order of rows and columns depends on a choice of generator and primitive root of unity.
The group of characters is isomorphic to G itself, but not naturally so, and is known as the dual group
,
in the language of Pontryagin duality
, and the original group G can be recovered as the double dual.
If an abelian group is expressed as a direct product, and the dual group likewise decomposed, and the elements of each sorted in lexicographic order, then the character table of the product group is the Kronecker product
(tensor product) of the character tables for the two component groups, which is just a statement that the value of a product homomorphism on a product group is the product of the values:
between ρ and τ is a linear map T : Cn → Cm so that for all g in G we have the following commuting relation: T ° ρ(g) = τ(g) ° T.
According to Schur's lemma
, a non-zero morphism between two irreducible complex representations is invertible, and moreover, is given in matrix form as a scalar multiple of the identity matrix.
This result holds as the complex numbers are algebraically closed. For a counterexample over the real numbers, consider the two dimensional irreducible real representation of the cyclic group
C4 = 〈x〉 given by:
Then the matrix
defines an automorphism of ρ, which is clearly not a scalar multiple of the identity matrix.
space Cn. It may turn out that Cn has an invariant subspace V ⊂ Cn. The action of G is given by complex matrices and this in turn defines a new representation σ : G → GL(V). We call σ a subrepresentation of ρ. A representation without subrepresentations is called irreducible.
to representation theory
.
s, a graphical method exists to determine their finite representations that associates with each representation a Young tableau
(also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
If ρ is an n-dimensional irreducible representation of G with the underlying vector space V, then we can define a G×G morphism of representations, for all g in G and x in V
where 1G is the trivial representation of G. This defines a G×G morphism of representations.
Now we use the above lemma and obtain the G×G morphism of representations
.
The dual representation of C[G] as a G×G-representation is equivalent to C[G]. An isomorphism is given if we define the contraction 〈g,h〉 = δgh. So, we end up with a G×G-morphism of representations
.
Then
for all x in
and y in V.
By Schur's lemma, the image
of f″ is a G×G irreducible representation, which is therefore n×n dimensional, which also happens to be a subrepresentation of C[G] (f″ is nonzero).
This is n direct sum equivalent copies V. Note that if ρ1 and ρ2 are equivalent G-irreducible representations, the respective images of the intertwining matrices would give rise to the same G×G-irreducible representation of C[G].
Here, we use the fact that if f is a function over G, then
We convert C[G] into a Hilbert space
by introducing the norm where 〈g,h〉 is 1 if g is h and zero otherwise. This is different from the 'contraction' given a couple of paragraphs back, in that this form is sesquilinear. This makes C[G] a unitary representation
of G×G. In particular, we now have the concepts of orthogonal complement and orthogonality of subrepresentations.
In particular, if C[G] contains two inequivalent irreducible G×G subrepresentations, then both subrepresentations are orthogonal to each other. To see this, note that for every subspace of a Hilbert space, there exists a unique linear transformation from the Hilbert space to itself which maps points on the subspace to itself while mapping points on its orthogonal complement to zero. This is called the projection map. The projection map associated with the first irreducible representation is an intertwiner. Restricted to the second irreducible representation, it gives an intertwiner from the second irreducible representation to the first. Using Schur's lemma, this must be zero.
Now suppose A ⊗ B is a G×G-irreducible representation of C[G].
This representation is also a G-representation (nA direct sum copies of B where nA is the dimension of A). If Y is an element of this representation (and hence also of C[G]) and X an element of its dual representation (which is a subrepresentation of the dual representation of C[G]), then
where e is the identity of G. Though the f″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G×G-irreducible subrepresentation of C[G], we have just showed that
as an irreducible G×G-subrepresentation of C[G] is contained in A ⊗ B as another irreducible G×G-subrepresentation of C[G]. Using Schur's lemma again, this means both irreducible representations are the same.
Putting all of this together,
There is a mapping from G to the complex numbers for each representation called the character
given by the trace of the linear transformation upon the representation generated by the element of G in question
All elements of G belonging to the same conjugacy class
have the same character: in other words χρ is a class function
on G. This follows from
by the cyclic property of the trace of a matrix.
What are the characters of C[G]? Using the property that gh−1 is only the same as g if h = e, χC[G](g) is |G| if g=e and 0 otherwise.
The character of a direct sum of representations is simply the sum of their individual characters.
Putting all of this together,
with the Kronecker delta on the right hand side.
Repeat this, working with characters of G×G instead of characters, of G which I'll call Δ. Then, ΔC[G](g,h) is the number of elements k in G satisfying g k h−1 = k. This is equal to
where * denotes complex conjugation. After all, C[G] is a unitary representation and any subrepresentation of a finite unitary representation is another unitary representation; and all irreducible representations are (equivalent to) a subrepresentation of C[G].
Consider
.
This is |G| times the number of elements which commute with g; which is |G|2 divided by the size of the conjugacy class of g, if g and k belong to the same conjugacy class, but zero otherwise. Therefore, for each conjugacy class Ci of size mi, the characters are the same for each element of the conjugacy class and so we can just call χρ(Ci) by an abuse of notation
). Then,
.
Note that
is a self-intertwiner (or invariant). This linear transformation, when applied to C[G] (as a representation of the second copy of G×G), would give as its image the 1-dimensional subrepresentation generated by
;
which is obviously the trivial representation
.
Since we know C[G] contains all irreducible representations up to equivalence and using Schur's lemma, we conclude that
for irreducible representations is zero if it's not the trivial irreducible representation; and it's of course |G|1 if the irreducible representation is trivial.
Given two irreducible representations Vi and Vj, we can construct a G-representation
,
this time not as a G×G representation but an ordinary G-representation. See direct product
of representations. Then,
.
It can be shown that any irreducible representation can be turned into a unitary irreducible representation. So, the direct product of two irreducible representations can also be turned into a unitary representations and now, we have the neat orthogonality property allowing us to decompose the direct product into a direct sum of irreducible representations (we're also using the property that for finite dimensional representations, if you keep taking proper subrepresentations, you'll hit an irreducible representation eventually. There's no infinite strictly decreasing sequence of positive integers). See Maschke's theorem
.
If i≠j, then this decomposition does not contain the trivial representation (Otherwise, we'd have a nonzero intertwiner from Vj to Vi contradicting Schur's lemma). If i=j, then it contains exactly one copy of the trivial representation (Schur's lemma states that if A and B are two intertwiners from Vi to itself, since they're both multiples of the identity, A and B are linearly dependent). Therefore,
Applying a result of linear algebra to both orthogonality relations (|Ci| is always positive), we find that the number of conjugacy classes is greater than or equal to the number of inequivalent irreducible representations; and also at the same time less than or equal to. The conclusion, then, is that the number of conjugacy classes of G is the same as the number of inequivalent irreducible representations of G.
We know that any irreducible representation can be turned into a unitary representation. It turns out the Hilbert space norm is unique up to multiplication by a positive number. To see this, note that the conjugate representation of the irreducible representation is equivalent to the dual irreducible representation with the Hilbert space norm acting as the intertwiner. Using Schur's lemma, all possible Hilbert space norms can only be a multiple of each other.
Let ρ be an irreducible representation of a finite group G on a vector space V of (finite) dimension n with character χ. It is a fact that χ(g) = n if and only if ρ(g) = id (see for instance Exercise 6.7 from Serre's book below). A consequence of this is that if χ is a non-trivial irreducible character of G such that χ(g) = χ(1) for some g≠1 then G contains a proper non-trivial normal subgroup
(the normal subgroup is the kernel
of ρ). Conversely, if G contains a proper non-trivial normal subgroup
N, then the composition of the natural surjective group homomorphism
G → G/N with the regular representation
of G/N produces a representation π of G which has kernel
N. Taking χ to be the character of some non-trivial subrepresentation of π, we have a character satisfying the hypothesis in the direct statement above. Altogether, whether or not G is simple
can be determined immediately by looking at the character table
of G.
of a finite group
G, over the complex number
s, were discovered by Ferdinand Georg Frobenius
in the years before 1900. Later the modular representation theory
of Richard Brauer
was developed.
extends many results about representations of finite groups to representations of compact groups.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, representation theory
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 studiesmodules over these abstract algebraic structures...
is a technique for analyzing abstract groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in terms of groups of linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s. See the article on group representations for an introduction. This article discusses the representation theory of groups that have a finite number of elements.
Basic definitions
All the linear representations in this article are finite dimensional and assumed to be complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
unless otherwise stated. A representation of G is a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
ρ:G → GL(n,C) from G to the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n 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 the inverse of an invertible matrix is invertible...
GL(n,C). Thus to specify a representation, we just assign a square matrix to each element of the group, in such a way that the matrices behave in the same way as the group elements when multiplied together.
We say that ρ is a real representation
Real representation
In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...
of G if the matrices are real. In other words if ρ(G) ⊂ GL(n,R).
Other formulations
A representation ρ: G → GL(n,C) defines a group actionGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of G on the vector space Cn. Moreover this action completely determines ρ. Hence to specify a representation it is enough to specify how it acts on its representing vector space.
Alternatively, the action of a group G on a complex vector space V
induces a left action of group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
C[G] on the vector space V, and vice-versa. Hence representations are equivalent to left C[G]-modules.
The group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
C[G] is a |G|-dimensional algebra over the complex numbers, on which G acts. (See Peter–Weyl for the case of compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
s.)
In fact C[G] is a representation for G×G. More specifically, if g1 and g2 are elements of G and h is an element of C[G] corresponding to the element h of G,
[h]=g1 h g2-1.
C[G] can also be considered as a representation of G in three different ways:
- Conjugation: g[h] = g h g−1
- As a left action: g[h] = g h (a regular representationRegular representationIn mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
) - As a right action: g[h] = h g−1 (also);
these are all to be 'found' inside the G×G action.
Example
For many groups it is entirely natural to represent the group through matrices.Consider for example the dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
D4 of symmetries of a square. This is generated by the two reflection matrices
Here m is a reflection that maps (x,y) to (− x,y), while n maps (x,y) to (y,x). Multiplying these matrices together creates a set of 8 matrices that form the group. As discussed above, we can either think of the representation in terms of the matrices, or in terms of the action on the two-dimensional vector space (x,y).
This representation is faithful - that is, there is a one-to-one correspondence between the matrices and the elements of the group. It is also irreducible, because there is no subspace of (x,y) that is invariant under the action of the group.
Discrete Fourier transform
If G is a finite cyclic group, then its representation theory is called the discrete Fourier transformDiscrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
; this example is central to digital signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
.
All irreducible representations are 1-dimensional (characters), and correspond to sending a generator of G to a root of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
, not necessarily primitive (the trivial representation sends a generator to 1, for instance).
A function on G is called the time domain representation of the function, while the corresponding expression in terms of characters is called the frequency domain representation of the function: changing from the time domain description to the frequency domain description is called the discrete Fourier transform, and the opposite direction is called the inverse discrete Fourier transform.
The character table, which in this case is the matrix of the transform, is the
DFT matrix
DFT matrix
A DFT matrix is an expression of a discrete Fourier transform as a matrix multiplication.-Definition:An N-point DFT is expressed as an N-by-N matrix multiplication as X = W x, where x is the original input signal, and X is the DFT of the signal.The transformation W of size N\times N can be defined...
, which is, up to normalization factor, the Vandermonde matrix for the nth roots of unity; the order of rows and columns depends on a choice of generator and primitive root of unity.
The group of characters is isomorphic to G itself, but not naturally so, and is known as the dual group
Dual group
In mathematics, the dual group may be:* The Pontryagin dual of a locally compact abelian group* The Langlands dual of a reductive algebraic group* The Deligne-Lusztig dual of a reductive group over a finite field....
,
in the language of Pontryagin dualityPontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
, and the original group G can be recovered as the double dual.
Abelian groups
More generally, any finite abelian group is a direct sum of finite cyclic groups (by the fundamental theorem of finitely generated abelian groups, though the decomposition is not unique in general), and thus the representation theory of finite abelian groups is completely described by that of finite cyclic groups, that is, by the discrete Fourier transform.If an abelian group is expressed as a direct product, and the dual group likewise decomposed, and the elements of each sorted in lexicographic order, then the character table of the product group is the Kronecker product
Kronecker product
In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...
(tensor product) of the character tables for the two component groups, which is just a statement that the value of a product homomorphism on a product group is the product of the values:
Morphisms between representations
Given two representations ρ: G → GL(n,C) and τ: G → GL(m,C) a morphismMorphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
between ρ and τ is a linear map T : Cn → Cm so that for all g in G we have the following commuting relation: T ° ρ(g) = τ(g) ° T.
According to Schur's lemma
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
, a non-zero morphism between two irreducible complex representations is invertible, and moreover, is given in matrix form as a scalar multiple of the identity matrix.
This result holds as the complex numbers are algebraically closed. For a counterexample over the real numbers, consider the two dimensional irreducible real representation of the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
C4 = 〈x〉 given by:
Then the matrix
defines an automorphism of ρ, which is clearly not a scalar multiple of the identity matrix.Subrepresentations and irreducible representations
As noted earlier, a representation ρ defines an action on a vectorspace Cn. It may turn out that Cn has an invariant subspace V ⊂ Cn. The action of G is given by complex matrices and this in turn defines a new representation σ : G → GL(V). We call σ a subrepresentation of ρ. A representation without subrepresentations is called irreducible.
Constructing new representations from old
There are number of ways to combine representations to obtain new representations. Each of these methods involves the application of a construction from linear algebraLinear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
to representation theory
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 studiesmodules over these abstract algebraic structures...
.
- Given two representations ρ1, ρ2 we may construct their direct sum ρ1 ⊕ ρ2 by (ρ1 ⊕ ρ2) (g)(v,w) = (ρ1(g)v, ρ2(g)w).
- The tensor representation of ρ1, ρ2 is defined by (ρ1 ⊗ ρ2) (v ⊗ w) = ρ1(v) ⊗ ρ2(w).
- Let ρ : G → GL(n,C) be a representation. Then ρ induces a representation ρ* on the dualDualDual may refer to:* Dual , a notion of paired concepts that mirror one another** Dual , a formalization of mathematical duality** . . ...
of the vector space Hom(Cn,C). Let f : Cn → C be a linear functional. The representation ρ* is then defined by the rule ρ* (g) (f) = f(ρ(g)−1). The representation ρ* is called either the dual representationDual representationIn mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation is defined over the dual vector space as follows:...
or the contragredient representation of ρ.
- Furthermore, if a representation ρ has a subrepresentation σ then the quotient of the representing vector spaces for ρ and σ has a well defined action of G on it. We call the resulting representation the quotient representation of ρ by σ.
Young tableau
For the symmetric groupSymmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
s, a graphical method exists to determine their finite representations that associates with each representation a Young tableau
Young tableau
In mathematics, a Young tableau is a combinatorial object useful in representation theory. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at...
(also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
Applying Schur's lemma
Lemma. If f: A ⊗ B → C is a morphism of representations, then the corresponding linear transformation obtained by dualizing B is: f′: A → C ⊗ B* is also a morphism of representations. Similarly, if g: A → B ⊗C is a morphism of representations, dualizing it will give another morphism of representations g′: A ⊗ C* → B.
If ρ is an n-dimensional irreducible representation of G with the underlying vector space V, then we can define a G×G morphism of representations, for all g in G and x in V
- f: C[G] ⊗ (1G ⊗ V) → (V ⊗ 1G)
- f:(g ⊗ x) = ρ(g)[x]
where 1G is the trivial representation of G. This defines a G×G morphism of representations.
Now we use the above lemma and obtain the G×G morphism of representations
.The dual representation of C[G] as a G×G-representation is equivalent to C[G]. An isomorphism is given if we define the contraction 〈g,h〉 = δgh. So, we end up with a G×G-morphism of representations
.Then
for all x in
and y in V.By Schur's lemma, the image
Image
An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...
of f″ is a G×G irreducible representation, which is therefore n×n dimensional, which also happens to be a subrepresentation of C[G] (f″ is nonzero).
This is n direct sum equivalent copies V. Note that if ρ1 and ρ2 are equivalent G-irreducible representations, the respective images of the intertwining matrices would give rise to the same G×G-irreducible representation of C[G].
Here, we use the fact that if f is a function over G, then
We convert C[G] into a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
by introducing the norm where 〈g,h〉 is 1 if g is h and zero otherwise. This is different from the 'contraction' given a couple of paragraphs back, in that this form is sesquilinear. This makes C[G] a unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
of G×G. In particular, we now have the concepts of orthogonal complement and orthogonality of subrepresentations.
In particular, if C[G] contains two inequivalent irreducible G×G subrepresentations, then both subrepresentations are orthogonal to each other. To see this, note that for every subspace of a Hilbert space, there exists a unique linear transformation from the Hilbert space to itself which maps points on the subspace to itself while mapping points on its orthogonal complement to zero. This is called the projection map. The projection map associated with the first irreducible representation is an intertwiner. Restricted to the second irreducible representation, it gives an intertwiner from the second irreducible representation to the first. Using Schur's lemma, this must be zero.
Now suppose A ⊗ B is a G×G-irreducible representation of C[G].
Note. The complex irreducible representations of G×H are always a direct product of a complex irreducible representation of G and a complex irreducible representation of H. This is not the case for real irreducible representations. As an example, there is a 2 dimensional real irreducible representation of the group C3 × C3 which transforms nontrivially under both copies of C3 but cannot be expressed as the direct product of two irreducible representations of C3.
This representation is also a G-representation (nA direct sum copies of B where nA is the dimension of A). If Y is an element of this representation (and hence also of C[G]) and X an element of its dual representation (which is a subrepresentation of the dual representation of C[G]), then
where e is the identity of G. Though the f″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G×G-irreducible subrepresentation of C[G], we have just showed that
as an irreducible G×G-subrepresentation of C[G] is contained in A ⊗ B as another irreducible G×G-subrepresentation of C[G]. Using Schur's lemma again, this means both irreducible representations are the same.Putting all of this together,
Theorem. C[G] ≅where the sum is taken over the inequivalent G-irreducible representations V.
Corollary. If there are p inequivalent G-irreducible representations, Vi, each of dimension ni, then |G| = n12 + ... + np2.
Character theory
- Main article: Character theoryCharacter theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
There is a mapping from G to the complex numbers for each representation called the character
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
given by the trace of the linear transformation upon the representation generated by the element of G in question
- χρ(g)=Tr[ρ(g)].
All elements of G belonging to the same conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
have the same character: in other words χρ is a class function
Class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G...
on G. This follows from
- Tr[ρ(ghg-1)]=Tr[ρ(g)ρ(h)ρ(g)-1]=Tr[ρ(h)]
by the cyclic property of the trace of a matrix.
What are the characters of C[G]? Using the property that gh−1 is only the same as g if h = e, χC[G](g) is |G| if g=e and 0 otherwise.
The character of a direct sum of representations is simply the sum of their individual characters.
Putting all of this together,
with the Kronecker delta on the right hand side.
Repeat this, working with characters of G×G instead of characters, of G which I'll call Δ. Then, ΔC[G](g,h) is the number of elements k in G satisfying g k h−1 = k. This is equal to
where * denotes complex conjugation. After all, C[G] is a unitary representation and any subrepresentation of a finite unitary representation is another unitary representation; and all irreducible representations are (equivalent to) a subrepresentation of C[G].
Consider
.This is |G| times the number of elements which commute with g; which is |G|2 divided by the size of the conjugacy class of g, if g and k belong to the same conjugacy class, but zero otherwise. Therefore, for each conjugacy class Ci of size mi, the characters are the same for each element of the conjugacy class and so we can just call χρ(Ci) by an abuse of notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
). Then,
.Note that
is a self-intertwiner (or invariant). This linear transformation, when applied to C[G] (as a representation of the second copy of G×G), would give as its image the 1-dimensional subrepresentation generated by
;which is obviously the trivial representation
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
.
Since we know C[G] contains all irreducible representations up to equivalence and using Schur's lemma, we conclude that
for irreducible representations is zero if it's not the trivial irreducible representation; and it's of course |G|1 if the irreducible representation is trivial.
Given two irreducible representations Vi and Vj, we can construct a G-representation
,this time not as a G×G representation but an ordinary G-representation. See direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of representations. Then,
.It can be shown that any irreducible representation can be turned into a unitary irreducible representation. So, the direct product of two irreducible representations can also be turned into a unitary representations and now, we have the neat orthogonality property allowing us to decompose the direct product into a direct sum of irreducible representations (we're also using the property that for finite dimensional representations, if you keep taking proper subrepresentations, you'll hit an irreducible representation eventually. There's no infinite strictly decreasing sequence of positive integers). See Maschke's theorem
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces...
.
If i≠j, then this decomposition does not contain the trivial representation (Otherwise, we'd have a nonzero intertwiner from Vj to Vi contradicting Schur's lemma). If i=j, then it contains exactly one copy of the trivial representation (Schur's lemma states that if A and B are two intertwiners from Vi to itself, since they're both multiples of the identity, A and B are linearly dependent). Therefore,
Applying a result of linear algebra to both orthogonality relations (|Ci| is always positive), we find that the number of conjugacy classes is greater than or equal to the number of inequivalent irreducible representations; and also at the same time less than or equal to. The conclusion, then, is that the number of conjugacy classes of G is the same as the number of inequivalent irreducible representations of G.
Corollary. If two representations have the same characters, then they are equivalent.
Proof. Characters can be thought of as elements of a q-dimensional vector space where q is the number of conjugacy classes. Using the orthogonality relations derived above, we find that the q characters for the q inequivalent irreducible representations forms a basis set. Also, according to Maschke's theorem, both representations can be expressed as the direct sum of irreducible representations. Since the character of the direct sum of representations is the sum of their characters, from linear algebra, we see they are equivalent.
We know that any irreducible representation can be turned into a unitary representation. It turns out the Hilbert space norm is unique up to multiplication by a positive number. To see this, note that the conjugate representation of the irreducible representation is equivalent to the dual irreducible representation with the Hilbert space norm acting as the intertwiner. Using Schur's lemma, all possible Hilbert space norms can only be a multiple of each other.
Let ρ be an irreducible representation of a finite group G on a vector space V of (finite) dimension n with character χ. It is a fact that χ(g) = n if and only if ρ(g) = id (see for instance Exercise 6.7 from Serre's book below). A consequence of this is that if χ is a non-trivial irreducible character of G such that χ(g) = χ(1) for some g≠1 then G contains a proper non-trivial normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
(the normal subgroup is the kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of ρ). Conversely, if G contains a proper non-trivial normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
N, then the composition of the natural surjective group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
G → G/N with the regular representation
Regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
of G/N produces a representation π of G which has kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
N. Taking χ to be the character of some non-trivial subrepresentation of π, we have a character satisfying the hypothesis in the direct statement above. Altogether, whether or not G is simple
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
can be determined immediately by looking at the character table
Character table
In group theory, a character table is a two-dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to classes of group elements...
of G.
History
The general features of the representation theoryGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
G, over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, were discovered by Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
in the years before 1900. Later the modular representation theory
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
of Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
was developed.
Generalizations
The Peter–Weyl theoremPeter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G...
extends many results about representations of finite groups to representations of compact groups.
See also
- Character theoryCharacter theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
- Real representationReal representationIn the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...
- Representation theory of the symmetric groupRepresentation theory of the symmetric groupIn mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum...
- Schur orthogonality relationsSchur orthogonality relationsIn mathematics, the Schur orthogonality relations express a central fact about representations of finite groups.They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as therotation group SO....
- Deligne–Lusztig theoryDeligne–Lusztig theoryIn mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by ....