Representation theory of finite groups

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In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of groups that have a finite number of elements.

Contents

1 Basic definitions
2 Other formulations
3 Morphisms between representations
4 Subrepresentations and irreducible representations
5 Constructing new representations from old
6 Applying Schur's lemma
7 Character theory
8 Examples
9 History
10 See also
11 References

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

[\rho:G\rightarrow GL(n,\mathbb)]

from G to the general linear group GL(n,C) is a group homomorphism, that is

[\rho(g h) = \rho(g) \rho(h)]

for any two elements [ g, h \in G].

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

[\rho:G\rightarrow GL(n,\mathbb)]

Other formulations

A representation [ \rho : G \rightarrow \mathrm(n, \mathbb) ] defines an group action of [ G ] on the vector space [ \mathbb^n ]. Moreover this action completely determines [ \rho ]. 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 [ \mathbb ] on the vector space V, and vice-versa. Hence representations are equivalent to left [ \mathbb ] 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 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,

(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.

Morphisms between representations

Given two representations [ \rho_1: G \rightarrow GL(n, \mathbb) ] and [ \rho_2: G \rightarrow GL(m, \mathbb) ] a morphism between [ \rho_1 ] and [ \rho_2 ] is a linear map

[ T : \mathbb^n \rightarrow \mathbb^m ]

so that for all [g] in [G]

[ T \circ \rho_1(g) = \rho_2(g) \circ 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 [ C_4 = \langle x \rangle ] given by

[ \rho : x \mapsto \begin 0 & 1 \\ -1 & 0 \end. ]

Then the matrix [ \begin 0 & 1 \\ -1 & 0 \end ] defines an automorphism of [ \rho ], which is clearly not a scalar multiple of the identity matrix.

Subrepresentations and irreducible representations

As noted earlier, a representation [ \rho ] defines an action on a vector space [ \mathbb^n ]. It may turn out that [ \mathbb^n ] has an invariant subspace [ V \subset \mathbb^n ]. The action of [ G ] is given by complex matrices and in this in turn defines a new representation [ \sigma : G \rightarrow\mathrm(V) ]. We call [ \sigma ] a subrepresentation of [ \rho ]. 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.

Given two representations [ \rho_1, \rho_2 ] we may construct their direct sum [ \rho_1 \oplus \rho_2 ] by

[ \rho_1 \oplus \rho_2(g)(v,w) = (\rho_1(g)v, \rho_2(g)w). ]

The tensor representation of [ \rho_1, \rho_2 ] is defined by

[ \rho_1 \otimes \rho_2(v \otimes w) = \rho_1(v) \otimes \rho_2(w) ].

Let [ \rho : G \rightarrow GL(n, \mathbb) ] be a representation. Then [ \rho ] induces a representation [ \rho^* ] on the dual of the vector space [ \mathrm( \mathbb^n, \mathbb) ]. Let [ f : \mathbb^n \rightarrow \mathbb ] be a linear functional. The representation [ \rho^* ] is then defined by the rule

[ \rho^*(g)f := f( \rho(g)^). ]

The representation [\rho^*] is called either the dual representation or the contragredient representation of [ \rho ].

Further more, if a representation [ \rho ] has an subrepresentation [ \sigma ] then the quotient of the representing vector spaces for [ \rho ] and [ \sigma ] has a well defined action of [ G ] on it. We call the resulting representation the quotient representation of [ \rho ] by [ \sigma ].

Applying Schur's lemma

Lemma: If

[f:A\otimes B\rightarrow C]

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

[f':A\rightarrow C\otimes \bar]

is also a morphism of representations. Similarly, if

[g:A\rightarrow B\otimes C]

is a morphism of representations, dualizing it will give another morphism of representations :[g':A\otimes \bar\rightarrow 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

[f:C[G]\otimes (1_G\otimes V)\rightarrow (V\otimes 1_G)]

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

[f(g\otimes x)=\rho(g)[x]]

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

[f':\bar\otimes V\rightarrow \overline].

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

=δ_gh,

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

[f'':\bar\otimes V\rightarrow C[G]].

It turns out

[f''(x\otimes y)=\sum_g]

for all x in [\bar] 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, 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

[\sum_f(g)hgk^=\sum_f(h^gk)g]

We convert C[G] into a Hilbert space by introducing the norm where 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 has got to be zero.

Now suppose [A\otimes B] 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

[f''(X\otimes Y)=\sum_g=\sum_g]

[=\sum_gY=\sum_(g,e)[Y]]

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 [A\otimes B] isn't a G-irreducible representation in general. But since [A\otimes B] 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 [\bar\otimes B] as an irreducible G×G-subrepresentation of C[G] is contained in [A\otimes 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,

[C[G]\simeq \sum_ \bar\otimes V.]

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

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

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,

[\sum_^p n_i \chi_(g)=|G|\delta_]

with the Kronecker delta on the RHS.

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

Then,

[\chi'_((g,h))]

is the number of elements in G satisfying

gkh^-1 = k.

This is equal to

[\sum_^p \chi__i}(g)\chi_(h)=\sum_^p \chi_(g^)\chi_(h)=\sum_^p \chi_(g)^*\chi_(h)]

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

[\sum_\chi'_((g,hkh^))].

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,

[\frac

>\delta_=\sum_^p\chi_(C_i)^*\chi_(C_j)].

Note that

[\sum_\rho(g)]

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

[\sum_g];

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

[\sum_\rho(g)]

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 :[\bar_i\otimes V_j],

this time not as a G×G representation but an ordinary G-representation. See direct product of representations. Then,

[\chi__i\otimes V_j}(g)=\chi__i}(g)\chi_(g)=\chi_(g)^*\chi_(g)].

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 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,

[\sum_\chi_(g)^*\chi_(g)=\sum_|C_k|\chi_(C_k)^*\chi_(C_k)=|G|\delta_]

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.

History

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.

References

The standard graduate level reference for representations of groups in general.

A beautiful and readable introduction; designed for self study.

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