In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. Later the modular representation theory of Richard Brauer was developed. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a finite group is a group which has finitely many elements. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... In mathematics, modular representation theory is the branch of representation theory that studies linear representations of finite group G over a field K such that the characteristic of K divides the order of G. In other words, the number of elements of G is zero when considered as an element... Richard Dagobert Brauer (February 10, 1901 - April 17, 1977) was a leading German and American mathematician. ...

Basic definitions

All the linear representations in this article are finite dimensional and assumed to be complex unless otherwise stated. We say that ρ is a representation of G if the mapping A complex is a whole that comprehends a number of parts, especially one with interconnected or mutually related parts. ...

from G to the general linear group GL(n,C) is a group homomorphism, that is In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... 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) · h(v) From this property, one can deduce that h maps the identity element...

ρ(gh) = ρ(g)ρ(h)

for any two elements .

We say that ρ is a real representation of G if the matrices are real:

Other formulations

A representation Failed to parse (unknown function math): rho : G rightarrow mathrm{GL}(n, math{bb}(C))

defines an group action of G on the vector space . Moreover this actio completely determines ρ. Hence to specify a representation it is enough to specify how it acts on its representing vector space. In mathematics, a symmetry group describes all symmetries of objects. ...

Alternatively, the action of a group G on a complex vector space V induces a left action of group algebra on the vector space V, and vice-versa. Hence representations are equivalent to left modules. In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ...

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 groups.) 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, In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ... 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. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ...

(g₁,g₂)[h]=g₁h g^-1₂.

C[G] can also be considered as a representation of G in three different ways:

• Conjugation: g[h]=ghg^-1
• As a left action: g[h]=gh (a regular representation)
• As a right action: g[h]=hg^-1 (also);

these are all to be 'found' inside the G×G action. 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. ...

Morphisms between representations

Given two representations and a morphism between ρ₁ and ρ₂ is a linear map In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

so that

.

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. In mathematics, Schurs lemma is now a generic term applied to theorems on the commutant of a module M that is simple. ...

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 given by In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... 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 a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

Then the matrix defines an automorphism of 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 vector space . It may turn out that has a invariant subspace . The action of G is given by complex matrices and in this in turn defines a new representation . 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 algebra to representation theory. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...

Given two representations ρ₁,ρ₂ we may construct their direct sum by

The tensor representation of ρ₁,ρ₂ is defined by

.

Let be a representation. Then ρ induces a representation ρ ^* on the dual of the vector space . Let be a linear functional. The representation ρ ^* is then defined by the rule Look up Dual in Wiktionary, the free dictionary A dual is a pair or a grouping of two. ...

ρ ^* (g)f: = f(ρ(g) ^{− 1}).

The representation $rho^*$ is called either the dual representation or the contragredient representation of ρ. If G is a group and ρ is a representation of it over the vector space V, then the dual representation is defined over the dual vector space as follows: is the transpose of ρ(g-1) for all g in G. is also a representation, as you may check explicitly. ... In 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: is the transpose of ρ(g−1) for all g in G. Then is also a representation, as may...

Further more, if a representation ρ has an 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.

Applying Schur's lemma

Lemma: If

is a morphism of representations, then the corresponding linear transformation obtained by dualizing B,

is also a morphism of representations. Similarly, if

is a morphism of representations, dualizing it will give another morphism of representations  :.

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

as follows where 1_G is the trivial representation of G.

for all g in G and x in V. This defines a G×G morphism of representations, as can be explicitly checked.

Do the dualization trick above 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>=δ_gh,

as you may check. So, we end up with a G×G-morphism of representations

.

It turns out

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). For images in Wikipedia, see Wikipedia:Images. ...

This, of course would be n direct sum equivalent copies V. Note that if ρ₁ and ρ₂ are equivalent G-irreducible representations, the respective images of the intertwiners 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 mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ... 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(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group...

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 has got to be zero. In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. X × Y = { (x, y) | x ∈ X and y...

Now suppose is a G×G-irreducible representation of C[G]. (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 C₃×C₃ which transforms nontrivially under both copies of C₃ which can't be expressed as the direct product of two Z₃ irreducible representations.) This representation is also a G-representation (n_A direct sum copies of B where n_A 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. I know the f′′ defined a couple of paragraphs back is only defined for G-irreducible representations and isn't a G-irreducible representation in general. But since is simply the direct sum of copies of B's and we've shown that each copy all maps to the same subG×G-irreducible representation of C[G], we've just showed that as an irreducible G×G-subrepresentation of C[G] is contained in 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,

Corollary: If there are p inequivalent G-irreducible representations, V_i, each of dimension n_i, then

|G| = Σ n_i².

Character theory

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 In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

χ_ρ(g)=Tr[ρ(g)].

All elements of G belonging to the same conjugacy class have the same character: in other words χ_ρ is a class function on G. This follows from 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. ... In mathematics, a class function in group theory 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. Such functions play a basic role in representation theory. ...

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, χ_C[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 RHS. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... RHS is a three letter acronym which can stand for: The Right-Hand Side of an equation The Royal Horticultural Society Rijnmond Hoogvliet Sport, an amateur football club from Rotterdam, the Netherlands. ...

Repeat this, working now with G×G characters this time instead of G characters, which I'll call χ′.

Then,

χ'_C[G]((g,h))

is the number of elements in G satisfying

gkh^-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 C_i of size m_i, the characters are the same for each element of the conjugacy class and so we can just call

χ_ρ(C_i)

by an abuse of notation). Then, 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 while being unlikely to introduce errors or cause confusion. ...

.

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. In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector space V over a field K and (ii) all elements g of G act on V as the identity mapping. ...

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 V_i and V_j, 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, In mathematics, one can often define a direct product of objects already known, giving a new one. ...

.

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. In mathematics, in particular group representation theory, Maschkes theorem is the basic result proving that linear representations of a finite group over the complex numbers break up into irreducible pieces. ...

If i ≠ j, then this decomposition does not contain the trivial representation (Otherwise, we'd have a nonzero intertwiner from V_j to V_i 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 V_i 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 (|C_i| 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.

Examples

See Representations of the symmetric group. In 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. ...

See also

• Real representation

In mathematics and theoretical physics, a real representation is a group representation that is equivalent to its complex conjugate and that also allows the matrices representing the group elements to be real — unlike a pseudoreal representation (symplectic representation). ...

References

• Fulton, William; and Harris, Joe (1991). Representation Theory: A First Course, New York: Springer. ISBN 0-387-97495-4. The standard graduate level reference for representations of groups in general.

• James, Gordon; and Liebeck, Martin (1993). Representations and Characters of Finite Groups, Cambridge: Cambridge University Press. ISBN 0-521-44590-6. A beautiful and readable introduction; designed for self study.