A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
REX CARTER BUTLER
1. Introduction
The purpose of this article is to introduce in a natural way the basics of finite
representation theory. After some preliminaries we will focus on ordinary representations, that is representations to vector spaces over the field C. This choice makes
the proofs much easier, since C is algebraically closed and is of characteristic 0.
It is noted that this article uses the classical ’Definition-Theorem-Corollary’ style
when appropriate but otherwise uses a more conversational approach. For the
purposes of this paper, assume all groups and summations are finite and all vector
spaces are of finite dimension. For examples, simply keep in mind your favorite
groups of order less than or equal 24.
2. Beginnings in Representation Theory
2.1. What is a Representation? The basic idea of representation theory is to
study groups by considering them as linear transformations of a vector space. This
is done as follows: given a group G and a vector space V we associate to each
g ∈ G a linear transformation ϕ(g) ∈ GL(V ) in a way that makes the function
ϕ : G → GL(V ) a group homomorphism. ϕ is then called a representation of G on
V.
Given such a representation ϕ, it immediately follows that ϕ(G) is a subgroup
of GL(V ), to which we can apply the ideas of linear algebra. If ϕ is injective, then
G∼
= ϕ(G), in which case ϕ is called a faithful representation.
2.2. FG-modules as Linear Group Actions. There are two ways to think of
representation theory: through the representations themselves, or through their
corresponding FG-modules, where F is a field, and G is a group. This is best
explained by analogy with another type of representation.
Let SA denote the group of permutations of a particular set A. Cayley’s Theorem
tells us any group G is isomorphic to a subgroup of SA for a suitably large set A.
Formally, this means we can construct an injective homomophism f : G → SA
so that G ∼
= f (G). As such, each concrete permutation f (g) ∈ SA ’acts like’
or ’represents’ the abstact group element g ∈ G. Also, if we have some other
homomorphism π : G → SA , not necessarily injective, the image of the group G
under π is a group of permutations isomorphic to some quotient group of G. Such
maps are called permutation representations of G.
Now the group G permutes the set A as follows: given π : G → SA define
· : G × A → A by setting g · a = π(g)a. By the properties of homomorphisms this
map has the following properties:
1·a=a
Date: March 24, 2004.
1
2
REX CARTER BUTLER
g · (h · a) = gh · a
These are exactly the axioms of a group action. We can also reverse the process
and define a permutation representation from a group action, as the reader should
verify. This establishes a bijective correspondence between permutation representations and group actions.
Analogously, if instead we are given a (linear) representation ϕ : G → GL(V )
and likewise define g · v = ϕ(g)v we again get the axioms of a group action plus two
requirements which say each g ∈ G must act linearly:
1·v =v
g · (h · v) = gh · v
g · (λv) = λ(g · v)
g · (v + w) = g · v + g · w
These four axioms define a ’linear group action’ or FG-module, where F is the
scalar field of V . In this case, we name the module Mϕ . As before, we can reverse
this construction: Given an FG-module M let ϕM (g)(v) = g · v. ϕM is then a
representation. The notation ϕM (g) will be used repeatedly later on, to emphasize
that we are talking about the linear transformation associated to g via M . Due
to the correspondence between representations and FG-modules we use definitions
for each interchangeably. For example, if ϕ is a faithful representation then ϕ has
trivial kernel, meaning
g · m = m for all m ∈ Mϕ implies g = 1
and vise versa, giving us a definition of faithful for FG-modules.
2.3. Linearizing a Group Action. Recall that the action of a linear transformation is completely determined by its action on any set of basis vectors. We can use
this fact to extend a group action to an FG-module, as follows:
Given that G acts on A, consider each element ai of A as a basis vector of a
vector space VA over a particular field F. We may construct this vector space VA
in one of the two following ways:
• Consider VA as the space of all functions from A to F, with each a ∈ A
associated to the function δa , where δa (x) is 1 if x = a and 0 otherwise.
These are the familiar Kronecker delta functions.
• Consider VA as the set of all formal sums of elements in A, where coefficients
are chosen from F.
We now define the action of G on VA via linear extension:
X
X
g·
λi ai =
λi (g · ai ).
i
i
An action of this type is called a permutation module for G. The matrices corresponding to each group element consist of 1’s and 0’s, with exactly one 1 in each
row and column. These are the aptly named permutation matrices.
2.4. The Group Algebra. Given a group G, any representation of G associates to
each group element a linear operator. Therefore, in the context of representation
theory, groups may be considered as abstract sets of linear operators. However,
groups are incomplete in this sense, as a ’complete’ set of linear operators will form
an algebra, i.e. a vector space with multiplication. We can make G into an algebra
as follows:
A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
3
Consider the vector space VG constructed as above. If we think of the elements
of the group G as invertible linear operators we can multiply elements of VG exactly
as we would multiply sums of linear transformations:
´
³X
´³ X
´ XX
X³X
λg g
µh h =
λg µh gh =
λh µh−1 g g.
g∈G
g∈G h∈G
h∈G
g∈G
h∈G
As VG is already a vector space, this definition of multiplication makes VG into an
algebra, which we denote by FG and name the Group Algebra of G.
Note that by linearity FG acts on any FG-module M just as
P G does, hence the
name ’FG-module’ [Note that FG is a ring]. Precisely, given g∈G λg g ∈ FG and
m ∈ M define:
³X
´
X
λg g · m =
λg (g · m)
g∈G
g∈G
Under this definition, any representation ϕ : G → GL(V ) extends to a unique
homomorphism of algebras ϕ̂ : FG → End(V ). This is all very natural, as GL(V )
is a subset of an algebra, not just a group.
Finally, just as a group acting on itself by left multiplication is an example of a
group action, we may consider the action of the ring FG on itself. This defines a
’ring action,’ or module, we name the Regular FG-module. Whenever we speak of
FG in module terms, this is what we mean.
Theorem 2.1. The Regular FG-module is faithful.
Proof. If g · m = m for all m ∈ FG then g · h = h for all h ∈ G, so g = 1.
¤
We now have F-linear versions of groups and group actions in the form of group
algebras FG and FG-modules.
2.5. Basic Terminology. The simplest example of a representation is defined as
follows. Define ϕ by ϕ(g) = 1V , the identity map on V where dim V = 1. Then
g · v = v for any choice of g and v. This is called the trivial representation or trivial
FG-module.
FG-modules, just like vector spaces, contain smaller substructures. V is a submodule of an FG-module W if V is a vector subspace of W that is invariant under
group action, so that g · V = V for all g ∈ G. If so, V is itself an FG-module whose
operations match those of W .
An FG-module M is said to be the direct sum of V and W if, as vector spaces,
M = V ⊕ W where V and W are submodules of M . As with vector spaces, we
can form external direct sums of FG-modules. Given FG-modules V and W , the
module direct sum V ⊕ W is the vector space direct sum V ⊕ W with group action
defined by g · (v + w) = g · v + g · w, where v ∈ V and w ∈ W . We will denote the
external direct sum M ⊕ M ⊕ . . . ⊕ M (where M occurs d times) by dM .
Let V be any FG-module with positive dimension. V is irreducible if it has
no submodules other than {0} and V . Otherwise V is reducible. V is completely
reducible if it can be expressed as a direct sum of irreducible submodules.
Various FG-modules are related by module homomorphisms. Given FG-modules
M and N , a linear transformation f : M → N is an FG-module homomorphism if it
’respects group action,’ i.e. if f (g · v) = g · f (v) for all g ∈ G. The homomorphisms
between M and N form a vector space homFG (M, N ).
A bijective module homomorphism ϕ is a called an isomorphism and indicates
that the modules concerned are of the same form [In this case, ϕ−1 will also be
4
REX CARTER BUTLER
a isomorphism of modules, as expected]. In terms of representations this means
the following: Remembering that f : M → N is linear and invertible, write it
as L : M → N . Then M is isomorphic to N if and only if L(g · v) = g · Lv
iff L(ϕM (g)v) = ϕN (g)Lv iff ϕM (g) = L−1 ϕN (g)L. This condition is called the
equivalence of the representations ϕM and ϕN .
FG-modules and FG-module homomorphisms form a category, and so when the
context is clear we sometimes just speak of ’modules’ and ’homomorphisms.’ As
with many algebraic systems, the kernel and image of an FG-module homomorpism
is an FG-module, as the reader should check.
2.6. Maschke’s Theorem. From now on we choose F = C. This assumption is
not required to prove the following theorem, but will be critical later.
Theorem 2.2. If V is a submodule of an CG-module M , then there exists a submodule W of M such that M = V ⊕ W . In this case, W is called the invariant
complement of V .
P
Proof. Let h , i be any inner product on M . Define (u, v) = h∈G < h · u, h · v >
Then ( , ) is a inner product for which (g · a, g · b) = (a, b) for all g ∈ G and
a,b ∈ M . In this case, V ⊥ is also an submodule of M , since it is invariant under
group action, as the previous inner product formula shows. Let W = V ⊥ .
¤
This immediately implies Maschke’s Theorem:
Theorem 2.3. Every CG-module M of positive dimension is completely reducible.
Proof. We apply strong induction on the dimension of M . If dim M = 1 then M
is irreducible. If dim M > 1 and M is not reducible it must contain an irreducible
module V . In this case, by the corollary above, M = V ⊕ W where dim V and
dim W are strictly less than dim M .
¤
This result tells us that all CG-modules are completely reducible. That is, given
any CG-module M ,
M
M=
Mi
i
where each Mi is irreducible submodule. There may, however, be multiple ways to
decompose a given module M into such a direct sum. For example, there are an
infinite number of ways to decompose any two dimensional CG-module whose group
acts trivially. However, all direct sum decompositions are equivalent in the sense
that, modulo isomorphism, any irreducible occurs as a summand a fixed number of
times. We will prove this later.
Maschke’s Theorem also allows us to decompose a module according to the kernel
and image of an exiting homomorphism.
Theorem 2.4. Let ϕ : M → N be a homomorphism of CG-modules. Then M =
ker ϕ ⊕ U where U is an isomorphic copy of im ϕ inside M .
Proof. By Mascke’s Theorem, M = ker ϕ ⊕ U for some module U . Define a new
homomorphism ϕ : U → im ϕ by restricting the domain of ϕ to U . Now each
m ∈ M is written uniquely as k + u with k ∈ ker ϕ and u ∈ U , where ϕ(k +
u) = ϕ(u) = ϕ(u). Therefore ϕ is onto, as it has the same range as ϕ. Also,
ker ϕ ∩ U = {0} so ker ϕ = {0}. Thus ϕ is also bijective.
¤
A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
5
2.7. Schur’s Lemma. As Mascke’s Theorem indicates, an understanding of the
’irreducibles’ is critical in the understanding of representations in general. This
begins with the following theorem, Schur’s Lemma, part 1 and part 2:
Theorem 2.5. If ϕ : M → N is a homomorphism of irreducible CG-modules, then
either ϕ is an isomorphism or the image of ϕ is {0}.
Proof. N is irreducible, so the image of ϕ is either {0} or N . In the second case,
the kernel of ϕ is a proper submodule of M , since not all of M is mapped to {0}.
Therefore ker ϕ = {0}, so ϕ is an isomorphism of vector spaces. It follows that ϕ
is bijective.
¤
Theorem 2.6. If ϕ : M → M is an endomorphism of an irreducible CG-module
then ϕ = λ1M for some scalar λ ∈ C.
Proof. ϕ is an operator on a complex vector space, which means it has an eigenvalue
λ. Therefore ker(ϕ − λ1M ) is a nonzero vector subspace of M . But ϕ − λ1M is
also an module automorphism, so ker(ϕ − λ1M ) is in fact a nontrivial submodule
of M . Since M is irreducible, ker(ϕ − λ1M ) = M , meaning ϕ = λ1M .
¤
Because of the first theorem, if M and N are nonisomorphic irreducible CGmodules then there is only one homomorphism from M to N , so
dim HomCG (M, N ) = 0.
On the other hand, if M and N are isomorphic irreducible CG-modules then
dim HomCG (M, N ) = 1.
We see this by equating M and N via an isomorphism ϕ. Then any homomorphism
ψ : M → N induces an endomorphism ϕ−1 ψ of M . But every endomorphism of M
acts like a scalar λ. As a result, ψ = λϕ. Thus ϕ spans HomCG (M, N ).
Though irreducible representations may have rich inner structure, the structure
of homomorphisms between them is trivial, as the previous discussion indicates.
2.8. Representations of Abelian Groups. Schur’s Lemma has important consequences for the irreducibles of an abelian group. Given an irreducible CG-module
M , where G is abelian, each x ∈ G acts like a module automorphism. That is, let
ϕx : M → M be defined by ϕx (m) = x · m. Now ϕx (g · m) = g · ϕx (m), so ϕx
is a module automorphism. This means any one dimensional subspace of M is
invariant under group action, as each ϕx acts like a scalar λx . Therefore M , being
irreducible, must have dimension 1. This proves:
Theorem 2.7. If M is an irreducible CG-module and G is abelian then dim M = 1.
It follows that every CH-module for abelian H decomposes completely into one
dimensional subspaces. This has the following useful corollary:
Theorem 2.8. Given a CG-module M and a designated element g ∈ G, we can
choose a basis so that the matrix of ϕM (g) is diagonal. In addition, the diagonal
entries of this matrix are nth roots of unity, where n is the order of g ∈ G.
Proof. Let H = hgi. We may reconsider M as a CH-module since CH ⊂ CG. In
this case M decomposes into the direct sum of one dimensional subspaces, since H
is abelian. Choosing a non-zero vector from each of these subspaces gives a vector
basis of M . The matrix of ϕM (g) under this basis is diagonal, as each submodule is
6
REX CARTER BUTLER
invariant under group action. Also, on each subspace, ϕM (g) must act like a scalar
of order n, meaning each diagonal entry must be an nth root of unity.
¤
2.9. The Irreducibles in the Group Algebra. The following two theorems
speak for themselves.
Theorem 2.9. Every irreducible CG-module M is isomorphic to a submodule of
CG.
Proof. Given an vector m ∈ M note that {r · m ∈ M : r ∈ CG} is a submodule of
M , since it is necessarily a vector subspace invariant under group action. Thus, if
m is non-zero, this set is all of M . Given such a vector, define ϕ : CG → M by
ϕ(r) = r · m. This defines a CG-module homomorphism which decomposes CG as
ker ϕ ⊕ U with U isomorphic to the image of ϕ, which is M .
¤
Theorem 2.10. There are only a finite number of non-isomophic irreducible CGmodules.
Proof. Write the group algebra CG as a finite direct sum of irreducibles Mi . If
M is an irreducible within CG, then at least one projection πj : CG → Mj does
not map all of M to zero. Now if we restrict the domain of πj to M , we find that
πj is a nontrivial homomorphism between irreducibles, making πj : M → Mj an
isomorphism.
¤
2.10. Direct Sums. We now discuss the properties of direct sums of CG-modules.
Recall from the theory of vector spaces that
M
M
XX
dim hom(
Vi ,
Wj ) =
dim hom(Vi , Wj ).
i
j
i
j
This exact equation holds if we instead consider CG-modules and module homomorphisms. To prove this, we show that as vector spaces,
M
M
MM
homCG (
Vi ,
Wj ) ∼
homCG (Vi , Wj ).
=
i
j
i
j
By induction, this equation holds in general if it holds when applied to the following
simple cases: homCG (V1 ⊕ V2 , W ) and homCG (V1 , W1 ⊕ W2 ). For these cases, the
proof is directly analogous to the equivalent equations in the theory of vector spaces.
With the equation above we can find the multiplicity
of each irreducible in the
L
group algebra. Consider any decomposition CG = i Mi where
P each Mi is an irreducible. Then, if M is any irreducible, dim homCG (CG, M ) = i dim homCG (Mi , M ).
But this is just a summation of one’s and zero’s according to whether or not
M ∼
= Mi . Therefore this summation is equal to the number of Mi ’s isomorphic
to M . This result also holds, by symmetry, for dim hom(M, CG). This proves:
Theorem 2.11. Given any irreducible CG-module M ,
dim homCG (CG, M ) = dim homCG (M, CG) = m
where m is the number of times M occurs isomorphically as a direct summand in
any complete decomposition of CG into irreducibles.
There is actually no requirement that CG be the particular direct sum considered
in the above theorem. (In fact, this is how we show the multiplicity of an irreducible
in any CG-module is the same under any decomposition.) However, in this case we
have another reduction:
A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
7
Theorem 2.12. For any CG-module M , dim homCG (CG, M ) = dim M .
Proof. Note any homomorphism ϕ : CG → M is completelyPdetermined by ϕ(1)
since by definition ϕ(r) = ϕ(r · 1) = r · ϕ(1). Express ϕ(1) as i λi bi where {bi }i∈I
is a vector basis of M . Now
X
ϕ(r) = r · ϕ(1) =
λi (r · bi ).
i
Choosing different ϕ results only in different coefficients λi , so the functions {r 7→
r · bi }i∈I span homCG (CG, M ). These functions are also linearly independent, for
if
X
0=
µi (r · bi )
i
then evaluation at r = 1 forces each µi to be zero. Therefore these functions form
a basis of size dimM .
¤
Combining the last two results tells us that isomorphic L
copies of an irreducible
Mi will appear exactly dimMi times in the decomposition i Mi of CG. Thus we
may reorder and collect the Mi by isomorphism type and prove the following:
Theorem 2.13. Let {V1 , V2 , . . . , Vk } be a complete set of non-isomorphic irreducibles. Then
M
ci Vi
CG ∼
=
i
where ci = dim Vi .
This decomposition places a strict restriction on the dimensions of irreducibles:
Theorem 2.14. Given a complete set of non-isomorphic irreducible CG-modules
Pk
2
{V1 , V2 , . . . , Vk } we have:
i=1 (dim Vi ) = |G|
L
Proof.
Let ci =PdimVi . The dimension of CG is |G| but it is also dim i ci Vi =
P
2
¤
i ci .
i dim ci Vi =
2.11. Decompositions of the Group Algebra. As an algebra, CG has an internal multiplication. On the other hand, as a module, CG may be decomposed in
various ways into a direct sum. If we multiply two elements a, b ∈ CG
Laccording to
, and let
suchP
a decomposition
the
result
is
not
very
instructive:
Let
CG
=
P
P i RiP
a = i ai , b = j bj where each ai ∈ Ri and bj ∈ Rj . Then ab = ( i ai )( j bj ) =
P P
i
j ai bj .
On the other hand, suppose we assume distinct Ri contain distinct irreducibles.
In this case, all homomorphisms from Ri to Rj will be trivial unless i = j [Decompose Ri and Rj into irreducibles and consider dim homCG (Ri , Rj )]. But any
element bj ∈ Rj induces an homomorphism ϕ : Ri → Rj defined by right multiplication: ϕ(ai ) = ai bj . The codomain is correct: Rj is a submodule of CG
so certainly ai bj ∈ Rj . It is also satisfies the homomorphism condition since
gϕ(ai ) = g(ai bj ) = (gai )bj = ϕ(gai ). Thus if i 6= j, ai bj = 0. This
P makes the
off diagonal entries in the double summation above zero, so ab =
i ai bi with
ai bi ∈ Ri . It also follows that rri , ri r ∈ Ri for all ri ∈ Ri , r ∈ CG. In other
P words,
each Ri is a two-sided ideal in the ring CG. If we represent 1 ∈ CG as i ei where
ei ∈ Ri , we find each ei is an idempotent two-sided unit for the subring Ri .
Summing up:
8
REX CARTER BUTLER
L
Theorem 2.15. As a module,
CG =
i Ri where
P assume P
P distinct Ri ’s contain
distinct irreducibles. Let 1 = i ei , a = i ai , and b = i bi where ei , ai , bi ∈ Ri .
(1) Each Ri is two-sided ideal in CG with unit ei .
(2) ei ej =P0 if i 6= j.
(3) ab = i ai bi .
2.12. The Center of the Group Algebra. Consider the center of the group
algebra:
Z(CG) = {z ∈ CG : zr = rz for all r ∈ CG}.
This is a vector subspace of CG.
Elements of the center of the group algebra are important because they act like
endomorphisms on any CG-module M . That is, given any z ∈ Z(CG), m 7→ z · m
is an endomorphism of M . Thus if M is irreducible, z must act like a scalar. It
follows that z also acts like a scalar on dM , for any d ≥ 0. In particular, if we
decompose the group algebra as
M
CG ∼
ci Vi
=
i
P
for certain scalars αi .
then z acts like a scalar αi on ci Vi . Thus z = i αi ei P
More precisely, given
any
z
∈
Z(CG),
write
1
=
i ei where each ei ∈ ci Vi .
P
P
P
α
e
for
certain
scalars αi . Therefore
ze
=
Then z = z1 = z( i ei ) =
i
i
i
i
i
Z(CG) is in the span of {ei }P
i∈I . In the opposite direction, it is easy to verify that
any expression of the form i αi ei is an element of Z(CG). Since there is one ei
for each irreducible, this proves:
Theorem 2.16. The dimension of Z(CG) is equal to the number of irreducible
CG-modules.
P
Suppose now we instead write z as z = g∈G λg g. In order for z to be in the
center of the group algebra, it must be that h1 zh = z for all h ∈ G. In other words
(by linearity)
X
X
λg h−1 gh =
λg g.
g∈G
g∈G
Matching coefficients gives λg = λh−1 gh , so the coefficients λg are constant on
conjugacy classes. This indicates a basis for Z(CG).
Given G, let C P
= {Ci }i∈I be the partition of G into conjugacy classes. Consider
the sums Ci+ =
g∈Ci g. These are linearly independent. Each of these sums
is invariant under conjugation since each Ci is a conjugacy class. It follows that
Ci+ g = gCi+ for all g ∈ G. Therefore, by linear extension, Ci+ ∈ Z(CG). Also,
any element of Z(CG) is in the span of these sums, as seen by the nature of the
coefficients λg above. This proves:
Theorem 2.17. The dimension of Z(CG) is equal to the number conjugacy classes
in G.
Combining our two theorems, we have:
Theorem 2.18. If a group G has k conjugacy classes there are exactly k irreducible
CG-modules.
A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
9
3. Characters
3.1. Defining the Character of a CG-module. An extremely important notion
in representation theory is that of a character. The character χ of a CG-module is
a function χ : G → C defined using the notion of trace.
First, a simple observation: We can define the trace of a linear operator just
as we can define the trace of a matrix. That is, given L : V → V define trace L
by trace ML where ML is the matrix of L under any basis. Since trace is invariant under conjugation, our definition is valid. We will use this property of trace
repeatedly.
We now define the character χ of a CG-module M as follows:
χ(g) = trace ϕM (g).
Though χ is usually thought of as being defined only on G, χ extends naturally to
all of CG, as all r ∈ CG ’name’ linear transformations of M . This linear extension
is natural and convenient, and lets us avoid switching between characters and traces
more often than necessary.
Since trace(B −1 AB) = trace(A), characters are invariant under conjugation:
χ(h−1 gh) = χ(g) for all g, h ∈ G.
Such functions are called class functions on G. By the same property of trace,
isomorphic CG-modules have equal characters, since M ∼
= N if and only if ϕM =
L−1 ϕN L for some invertible L : M → N .
If χ is the character of some CG module M , χ is called a character of G. Characters are particularly nice invariants because of the following.
Theorem
the character of M and χi the character of Mi .
L 3.1. Let χ be P
If M = i Mi then χ = i χi .
Proof. This is essentially a consequence
of the following fact: Given linear maps
P
Li : Mi → Mi , trace ⊕i Li = i trace Li . In matrix form, this means the trace
of a block diagonal matrix is the sum of the traces of the diagonal submatrices.
And now, for the proof: the character of M evaluated at g ∈ G is the trace of
ϕM (g) = ⊕i ϕMi (g).PGiven any g P
∈ G, let L = ϕM (g) and Li = ϕMi (g). Then
¤
χ(g) = trace ⊕i L = i trace Li = i χi (g).
This theorem has the following particularly enlightening corollary.
Theorem 3.2. Let {V1 , V2 , . . . , Vk } be a complete set of non-isomorphic irreducible
CG-modules, and let χi be the character of Vi .
Then, given any CG-module M with character ω,
X
M
ω=
ci χi where M =
ci Vi .
i
i
Thus characters ’mirror’ representations... CG-modules are direct sums of irreducible CG-modules, and characters are sums of corresponding irreducible characters.
Theorem 3.3. Given a CG-module M with character χ:
(1) χ(1) = dim M .
(2) χ(g) is a sum of mth roots of unity, where m is the order of g.
(3) χ(g −1 ) = χ(g).
10
REX CARTER BUTLER
Proof.
(1) The matrix of the identity operator on M has dim M ones on the diagonal.
(2) Above we proved that we can pick a basis of M so the matrix of ϕM (g) is
diagonal, with each diagonal entry an mth root of unity.
(3) Choose a basis of M so the matrix A of ϕM (g) is diagonal, as in part two
above. Let λ1 , λ2 , . . . , λn be the diagonal elements of a matrix A. Then
X
X
X
trace A−1 =
λ−1
=
λi = trace A
λi =
i
i
i
i
¤
3.2. Characters, Class Functions, and Inner Products. Recall that the characters of G are class functions on G. Consider the set all such class functions:
Class(G) = {f : G → C | f (h−1 gh) = f (g) for all g, h ∈ G}.
This set forms a vector space over C by defining operations pointwise.
Theorem 3.4. The dimension of Class(G) is equal to the number of G-conjugacy
classes.
Proof. For each G-conjugacy class Ci , define a fi : G → C by fi (g) = 1 if g ∈ Ci
and fi (g) = 0 if g ∈ G \ Ci . These functions fi are linearly independent and they
span Class(G). Therefore the dimension of Class(G) is equal to the number of fi
which is equal to the number of conjugacy classes Ci .
¤
Theorem 3.5. If the irreducible characters of G are linearly independent, they
span Class(G).
Proof. If the irreducible characters are linearly independent, they span a subspace
of Class(G) equal to the number k of irreducible CG-modules. But recall that k
is exactly equal to the number of G-conjugacy classes, which is the dimension of
Class(G).
¤
Our goal now is to show that the irreducble characters of G are linearly independent. We will do so in the next section, using an inner product structure that
we now define.
Given two class functions χ and ψ define:
1 X
hχ, ψi =
χ(g)ψ(g)
|G|
g∈G
Because we can replace complex conjugacy by inversion, two characters (or class
functions) χ and ψ play symmetric roles in the definition of hχ, ψi, so
hχ, ψi = hψ, χi.
3.3. Characters and the Group Algebra.
Theorem 3.6. Let χ be the character of the group algebra CG.
• χ(1) = |G|.
• χ(g) = 0 for g 6= 1.
Proof. Fix G as the basis of CG. Note the standard action (g, x) 7→ gx of G on
itself is fixed-point free unless g = 1. In module terms, this means the matrix of
ϕCG (g) has zeros along the main diagonal unless g = 1. Therefore χ(g) = 0 for
g 6= 1 and χ(1) = dimCG = |G| otherwise.
¤
A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
11
For the rest of the section, assume the following: Let CG = R1 ⊕ R2 where
R1 and R2 share no common irreducible components. Let χ,χ1 , and χ2 be the
characters of CG, R1 , and R2 respectively.
Now 1 ∈ CG is the sum of e1 ∈ R1 and e2 ∈ R2 . For certain coefficients λg ∈ C,
X
e1 =
λg g.
g∈G
We now find the coefficients λg in terms of χ1 . Recall that we have defined characters for the whole group algebra CG, not just G. Given x ∈ G, consider the value
of χ(e1 x−1 ). Since e1 acts like one on R1 and zero on R2 ,
χ(e1 x−1 ) = χ1 (x−1 ) + χ2 (0) = χ1 (x−1 )
P
−1
But also χ(e1 x−1 ) =
), and this is a summation of λg |G| when
g∈G λg χ(gx
g = x and 0 otherwise. Therefore
χ(e1 x−1 ) = λx |G|.
By solving, we find that
λx =
In other words,
e1 =
χ1 (x−1 )
.
|G|
1 X
χ1 (g −1 )g.
|G|
g∈G
Now we know e1 = e21 , so the coefficient of 1 in e1 equals the coefficient of 1 in
2
e1 . The former is just χ(1)
|G| . The latter is the coefficient of 1 in the product
´³ 1 X
´
³ 1 X
χ1 (g −1 )g
χ1 (h−1 )h
e1 e1 =
|G|
|G|
g∈G
which is
h∈H
1 X
|G|
2
χ1 (g)χ1 (g −1 ).
g∈G
−1
If we replace χ(g ) with χ(g) we see this number is just
first and second numbers tells us that
1
|G| hχ1 , χ1 i.
Equating the
hχ1 , χ1 i = χ1 (1).
We use this critical fact twice in the following proof.
Theorem 3.7. Let U and V be irreducible CG-modules with characters ψ and ω.
If U and V are isomorphic, then hψ, ωi = 1. Otherwise hψ, ωi = 0.
L
Proof. First, choose any decomposition of CG as
Wi where the Wi are irreducible. If U and V are isomorphic, let R1 be the direct sum of all dim U irreducible
copies of U in {Wi }i∈I . Let R2 be the direct sum of the remaining irreducibles, so
that CG = R1 ⊕ R2 . In this case, ψ will be the character of both U and V . If χ1
is the character of R1 , χ1 = dψ with d = dim U , so
hdψ, dψi = dψ(1) = d2
and it follows that hψ, ψi = 1.
On the other hand, if U and V are distinct, let R1 be the direct sum all of
irreducibles in {Wi }i∈I isomorphic to either U or V . Choose R2 so that CG =
12
REX CARTER BUTLER
R1 ⊕ R2 . Again, let χ1 be the character of R1 . In this case, χ1 = mψ + nω where
m = dimU and n = dimV . First, note χ1 (1) = mψ(1) + nω(1) = m2 + n2 . But
since χ1 (1) = hχ1 , χ1 i,
mψ(1) + nω(1) =
hmψ + nω, mψ + nωi =
m2 hψ, ψi + mnhψ, ωi + nmhω, ψi + n2 hω, ωi =
m2 + n2 + 2mn(hψ, ωi).
Therefore hψ, ωi = 0.
¤
3.4. The Irreducible Characters. For the rest of this section, assume the following: Let {V1 , V2 , . . . , Vk } be a complete set of non-isomorphic irreducible CGmodules. Let {χ1 , χ2 , . . . , χk } be the irreducible characters of a group G, so that χi
is the character of Vi . Note that k denotes the number of non-isomorphic irreducible
CG-modules. Immediately, by the previous section, we have:
Theorem 3.8. The irreducible characters of G are orthonormal. In other words:
hχi , χj i = δij
In particular, it follows that the irreducible characters χi are linearly independent. Therefore they span a subspace of dimension k inside Class(G). But we have
proved that k is the number of G-conjugacy classes, which is exactly the dimension
of Class(G). Therefore:
Theorem 3.9. The irreducible characters of G form an orthonormal basis for the
class function on G. In particular, if G is abelian, the characters of G span all
functions f : G → C. This is a finite analogue of Fourier decomposistion.
We can now use the inner product of characters to determine the multiplicity
of an irreducible in a CG-module. This leads to the following amazing result:
CG-modules are isomorphic if and only if their characters are equal.
Theorem
L3.10. If M is any CG-module with character ω, for certain coefficients
di , M ∼
= i di Vi . In this case:
P
(1) ω = i di χi
(2) dk = hχk ,P
ωi
(3) hω, ωi = i di 2
(4) If hω, ωi = 1 then M is irreducible.
(5) Any CG-module with character ω is isomorphic to M . In other words,
characters fully determine representations.
Proof.
(1) We proved this above.
(2) hχi , P
ωi =
hχi , j dj χj i =
P
d hχ , χ i =
Pj j i j
j dj δij = di
(3) hω,
P ωi = P
h i di χi , j dj χj i =
P P
d d hχ , χ i =
Pi Pj i j i j
dd δ =
Pi 2 j i j ij
d
i i
A QUICK INTRODUCTION TO FINITE REPRESENTATIONS
13
P
(4) If hω, ωi = i di 2 = 1 then exactly one di is equal to one, and the rest are
zero. Therefore M is irreducible.
L
(5) Let N be a module with character ω. For certain ci , N ∼
=
i ci Vi . But
ci = hχi , ωi = di , so N ∼
= M . f In short, the character of a CG-module
determines the multiplicity of any irreducible within it.
¤
Theorem 3.11. Let ψ and ω be the characters of V and W . Then
dim homCG (V, W ) = hψ, ωi
This number is called the intertwining number of V and W .
L
L
P
Proof.
i ci Vi and W as
i di Vi . In this case ψ =
i ci χi and
P Write V as
ω = i di χi . Now
X
X
hψ, ωi =h
ci χi ,
dj χj i
i
=
=
=
XX
i
j
i
j
XX
X
j
ci dj hχj , χl i
ci dj δij
ci di .
i
But
M
M
XX
dim homCG (
ci Vi ,
d j Vj ) =
dim homCG (ci Vi , dj Vj )
i
j
=
=
i
j
i
j
XX
X
δij ci dj
ci di
i
since dim homCG (Vi , Vj ) = δij .
¤
14
REX CARTER BUTLER
References
[1] J.L. Alperin and Rowen B. Bell. Groups and Representations. Springer, New York, 1995.
[2] David S. Dummit and Richard Foote. Abstract Algebra. John Wiley & Sons, New York, 1999.
[3] Gordon James and Martin Liebeck. Representations and Characters of Groups. Cambrige
University Press, Cambrige, 1993.
E-mail address: RexButler@hotmail.com
Department of Mathematics – Undergraduate, The University Of Utah, Salt Lake
City, Utah.