This document provides an overview of character theory for finite groups and its analogy to representation theory for the infinite compact group S1. It discusses several key concepts:
1) Character theory characterizes representations of a finite group G by their characters, which are class functions that map G to complex numbers. Characters determine representations up to isomorphism.
2) The irreducible representations of S1 are 1-dimensional and in bijection with integers n, where each representation maps z to zn.
3) By analogy to Fourier analysis, the characters of S1 form an orthonormal basis for L2(S1) and decompose representations into irreducibles in the same way as for finite groups.
Brian Covello: Research in Mathematical Group Representation Theory and SymmetryBrian Covello
Brian Covello's research review on group representation theory and symmetry. 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. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, 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.
Brian Covello: Research in Mathematical Group Representation Theory and SymmetryBrian Covello
Brian Covello's research review on group representation theory and symmetry. 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. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, 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.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
Dirigido a niños de 3-4 años.
Audio incorporado para facilitar la lectura autónoma.
Contiene información básica de hábitos de higiene, así como unas adivinanzas.
Esta presentacion muestra el programa de enfermeria de la universidad de ciencias aplicadas y ambientales udca, como lo son la vision, la mision, lo relacionado con la facultad, los objetivos, caracteristicas generales de la facultad de enfermeria
informe alternativo sobre pueblos indígenas y afrodescendientesCrónicas del despojo
- IDEHPUCP envía informe alternativo para el Comité para la Eliminación de la Discriminación Racial de las Naciones Unidas (CERD).
Servindi, 14 de octubre, 2016.- El Instituto de Democracia y Derechos Humanos de la Pontificia Universidad Católica del Perú (IDEHPUCP).envió un informe alternativo al Comité para la Eliminación de la Discriminación Racial (CERD).
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
Section 18.3-19.1.Today we will discuss finite-dimensional.docxkenjordan97598
Section 18.3-19.1.
Today we will discuss finite-dimensional associative algebras and their representations.
Definition 1. Let A be a finite-dimensional associative algebra over a field F . An element
a ∈ A is nilpotent if an = 0 for some positive integer n. An algebra is said to be nilpotent if
all of its elements are.
Exercise 2. Every subalgebra and factor algebra of a nilpotent algebra are nilpotent. Con-
versely, if I ⊂ A is a nilpotent ideal, and the quotient algebra A/I is nilpotent, then so is
A.
Example. The algebra of all strictly lower- (or upper-) triangular n×n matrices is nilpotent.
Proposition 3. If the algebra A is nilpotent, then An = 0 for some n ∈ Z+, that is the
product of any n elements if the algebra A equals 0.
Proof. Let B ⊂ A be the maximal subspace for which there exists n ∈ Z+ such that Bn = 0.
Note that B is closed under multiplication, i.e. B ⊃ Bk for any k ∈ Z+. Assume B 6= A
and choose an element a ∈ A\B. Since aBn = 0, there exists k ∈ Z+ such that aBk 6⊂ B.
Replacing a with a non-zero element in aBk we obtain aB ⊂ B. Recall that there exists
m ∈ Z+ so that am = 0. Now, let us set C = B ⊕〈a〉. Then we have
Cmn = 0
which contradicts the definition of the subspace B. �
Unless A is commutative, the set of all nilpotent elements of A does not have to be an
ideal (in general, not even a subspace). On the other hand, if I,J are nilpotent ideals in A,
then so is
I + J = {x + y |x ∈ I,y ∈ J} .
Therefore, there exists the maximal nilpotent ideal which contains every other nilpotent ideal
of A.
Definition 4. The radical of A is the maximal nilpotent ideal of A it is denoted rad(A).
The algebra A is semisimple if rad(A) = 0.
If char(F) = 0, there exists an alternative description of semisimple algebras. Consider
the regular representation of the algebra A
ρ: A → L(A), ρ(a)(b) = ab,
and define a “scalar product” on A via
(a,b) = tr(ρ(ab)) = tr(ρ(a)ρ(b)).
One can see that (·, ·) is a symmetric bilinear function on A satisfying (ab,c) = (a,bc).
Definition 5. For any ideal I ⊂ A, its orthogonal complement I⊥ is defined as
I⊥ = {a ∈ A |(a,i) = 0 for all i ∈ I} .
Proposition 6. For any ideal I ⊂ A, its orthogonal complement I⊥ ⊂ A is also an ideal.
Proof. For any a ∈ A, x ∈ I⊥, and y ∈ I we have
(xa,y) = (x,ay) = 0 and (ax,y) = (y,ax) = (ya,x) = 0.
�
1
2
Proposition 7. If A is an algebra over a field F with char(F) = 0, then every element a ∈ A
orthogonal to all of its powers is nilpotent.
Proof. Let a ∈ A be such that
(a,an) = tr ρ(a)n+1 = 0 for all n ∈ Z+.
Let L ⊃ F be the splitting field of the characteristic polynomial f(x) of the operator ρ(a).
Then, over L we have
f(x) = tk0
s∏
i=1
(t−λi)ki,
where λi are distinct for i = 1, . . . ,s, and
tr ρ(a)n+1 =
s∑
i=1
kiλ
n+1
i = 0.
Letting n run through the set 1, . . . ,s, the above equation yields a system of s homogeneous
linear equations in variables k1, . . . ,ks. The determinant of this system equals
λ21 . . .λ
2
nV (λ1, . . . ,λn),
where V (λ1.
Section 18.3-19.1.Today we will discuss finite-dimensional.docxrtodd280
Section 18.3-19.1.
Today we will discuss finite-dimensional associative algebras and their representations.
Definition 1. Let A be a finite-dimensional associative algebra over a field F . An element
a ∈ A is nilpotent if an = 0 for some positive integer n. An algebra is said to be nilpotent if
all of its elements are.
Exercise 2. Every subalgebra and factor algebra of a nilpotent algebra are nilpotent. Con-
versely, if I ⊂ A is a nilpotent ideal, and the quotient algebra A/I is nilpotent, then so is
A.
Example. The algebra of all strictly lower- (or upper-) triangular n×n matrices is nilpotent.
Proposition 3. If the algebra A is nilpotent, then An = 0 for some n ∈ Z+, that is the
product of any n elements if the algebra A equals 0.
Proof. Let B ⊂ A be the maximal subspace for which there exists n ∈ Z+ such that Bn = 0.
Note that B is closed under multiplication, i.e. B ⊃ Bk for any k ∈ Z+. Assume B 6= A
and choose an element a ∈ A\B. Since aBn = 0, there exists k ∈ Z+ such that aBk 6⊂ B.
Replacing a with a non-zero element in aBk we obtain aB ⊂ B. Recall that there exists
m ∈ Z+ so that am = 0. Now, let us set C = B ⊕〈a〉. Then we have
Cmn = 0
which contradicts the definition of the subspace B. �
Unless A is commutative, the set of all nilpotent elements of A does not have to be an
ideal (in general, not even a subspace). On the other hand, if I,J are nilpotent ideals in A,
then so is
I + J = {x + y |x ∈ I,y ∈ J} .
Therefore, there exists the maximal nilpotent ideal which contains every other nilpotent ideal
of A.
Definition 4. The radical of A is the maximal nilpotent ideal of A it is denoted rad(A).
The algebra A is semisimple if rad(A) = 0.
If char(F) = 0, there exists an alternative description of semisimple algebras. Consider
the regular representation of the algebra A
ρ: A → L(A), ρ(a)(b) = ab,
and define a “scalar product” on A via
(a,b) = tr(ρ(ab)) = tr(ρ(a)ρ(b)).
One can see that (·, ·) is a symmetric bilinear function on A satisfying (ab,c) = (a,bc).
Definition 5. For any ideal I ⊂ A, its orthogonal complement I⊥ is defined as
I⊥ = {a ∈ A |(a,i) = 0 for all i ∈ I} .
Proposition 6. For any ideal I ⊂ A, its orthogonal complement I⊥ ⊂ A is also an ideal.
Proof. For any a ∈ A, x ∈ I⊥, and y ∈ I we have
(xa,y) = (x,ay) = 0 and (ax,y) = (y,ax) = (ya,x) = 0.
�
1
2
Proposition 7. If A is an algebra over a field F with char(F) = 0, then every element a ∈ A
orthogonal to all of its powers is nilpotent.
Proof. Let a ∈ A be such that
(a,an) = tr ρ(a)n+1 = 0 for all n ∈ Z+.
Let L ⊃ F be the splitting field of the characteristic polynomial f(x) of the operator ρ(a).
Then, over L we have
f(x) = tk0
s∏
i=1
(t−λi)ki,
where λi are distinct for i = 1, . . . ,s, and
tr ρ(a)n+1 =
s∑
i=1
kiλ
n+1
i = 0.
Letting n run through the set 1, . . . ,s, the above equation yields a system of s homogeneous
linear equations in variables k1, . . . ,ks. The determinant of this system equals
λ21 . . .λ
2
nV (λ1, . . . ,λn),
where V (λ1.
1. A NOTE ON CHARACTER THEORY
FAN HUANG
In this note, we explore two examples. We try to glimpse both the analogy between
character theory of finite groups and representation theory of the infinite compact group,
and the power of character theory for nonabelian groups.
1. Basic Facts of Character Theory of Finite Groups
Let G be a finite group and ρ : G → GL(V ) be a linear representation of G in the complex
finite dimensional vector space V . For each g ∈ G, put χρ(g) = Tr(ρ(g)). The complex
valued function χρ on G is called the character of the representation ρ and it is a class
function– those are any functions from G into F which are constant on the conjugacy classes
of G, i.e., f : G → F such that f(g−1
xg) = f(x) for all g, x ∈ G. It characterizes the
representation ρ in the sense that representations are determined by their characters, up to
isomorphisms. Let
ρ : G → GL(V )
τ : G → GL(W)
be two representations of a group G on vector spaces V and W. An isomorphism from V
and W is a G-linear map f : V → W such that
τ(g) ◦ f(v) = f ◦ ρ(g)(v), ∀g ∈ G, ∀v ∈ V
i.e., for all g ∈ G, the below square commutes.
V V
W W
ρ(g)
f
τ(g)
f
In that case, we say that two representations ρ and τ are isomorphic. In terms of matrices,
two matrix representations are isomorphic if and only if they describe the same representation
in different bases. Specifically, choose a basis A = {a1, . . . , an} for V and a basis B =
{b1, . . . , bn} for W, and let
ρA : G → GLn(C)
τB : G → GLn(C)
be the two matrix representations obtained by writing ρ and τ in these bases. Then ρ and
τ are isomorphic if and only if ρA and τB are isomorphic.
Proof. It can be illustrated via the commutative graphs:
Date: August 2013.
1
2. A NOTE ON CHARACTER THEORY 2
Cn
Cn
W W
V V
Cn
Cn
τB(g)
B B
f f
A A
τ(g)
ρ(g)
ρA(g)
S S−1
There is an inner product on the vector space of class functions given by
f, h =
1
|G| g∈G
f(g)h(g),
where |G| is the order of the group G. The characters of irreducible representations form an
orthonormal basis for the space of class functions on G, denoted as Cl(G). Every represen-
tation is a direct sum of irreducible representations. Let ρ be any representation of G and ρi
be an isomorphism class of irreducible representations of G. Let χ and χi be the characters
of ρ and ρi respectively. Then the multiplicity of ρi in ρ is the inner product χi, χ .
2. The representation theory of the circle group S1
(the simplest infinite compact group)
Consider the unit circle S1
= {z ∈ C×
| |z|= 1} = {(x, y) ∈ R2
| x2
+ y2
= 1} in the
complex plane. If we view the circle as the set of points {eiθ
: θ ∈ R}, the natural group
operation is multiplication of complex numbers. We treat S1
as an abelian topological group
under multiplication with its topology as a subset of C. (A topological group is a group G
equipped with a topology such that the product map G×G → G and the inverse map G → G
are both continuous. More generally, a representation of a topological group on a vector
space V is a continuous group homomorphism ρ : G → GL(V ) with the topology of GL(V )
inherited from the space End(V ) of continuous linear operators). However, if we identify
points on S1
with their angle θ, then S1
becomes R modulo 2π, denote it R/2πZ, where the
operation is addition.The group isomorphism R/2πZ → S1
is given by t → eit
= cos t+i sin t.
Math 597 Fan Huang
3. FAN HUANG 3
From [JM74], since it is a continuous bijection between compact topological spaces, it is a
homeomorphism.
In this note we will focus on unitary representations of S1
, i.e., representations on a
Hilbert space which preserve the inner product. Unitary representations are those for which
the operators ρg are unitary, i.e. preserve the inner product. On the complex plane, this
means ρgv, ρgw = v, w = wv, ∀v, w ∈ V, g ∈ G. which means equivalently that ρg
t
ρg = I.
We denote the unitary representation of S1
as ρn : S1
→ U(V ).
It is not hard to see that for each integer n, the formula ρn(z) = zn
gives a 1-dimensional
irreducible representation of S1
. We shall show that
(1) Every irreducible unitary representation of S1
has dimension 1.
(2) Every continuous irreducible representation of S1
is of this form.
(3) For n = k, ρn and ρk are not isomorphic.
That is, Z is isomorphic to irreducible representations of S1
with bijection given by n → ρn.
First, we note that if W is an irreducible unitary representation of S1
, then dim W = 1.
To see this, we take as a fact that ρ is determined by ρ(g) where g ∈ S1
is a topological
generator. Then by density, we know where every other element goes. The spectral theorem
from linear algebra asserts that any unitary transformation is diagonalizable. Put another
way, ρ (g) = Sρ(g)S−1
, where S ∈ U(V ) and ρ (g) is a diagonal matrix. The eigenvalues of
unitary matrices all have absolute value 1. It is obvious that for each i the diagonal term
ρi(g) is a 1-dimensional unitary irreducible representation. This follows from ρ = n
i=1 ρi,
ρ = n
i=1 SρiS−1
and thus dim W = 1 as desired.
Proposition 2.1. Every irreducible unitary representations of S1
are the continuous homo-
morphisms of the form z → zn
for some n ∈ Z.
To prove this proposition, we need two Lemmas.
R R
S1
S1
φ
ψ
ρ
ψ
Lemma 2.1. Consider (R, +). If φ : R → R is a continuous homomorphism, then φ is a
multiplication by a scalar.
Proof. Set c = φ(1). Then φ(n) = nc, n ∈ Z. Also mφ( 1
m
) = c and then φ( 1
m
) = c
m
, m ∈ Z.
Thus φ( n
m
) = c n
m
and so φ(x) = cx, for x ∈ Q. Since Q is dense in R and φ is continuous,
we have φ(x) = cx for all x ∈ R.
Lemma 2.2. If ψ : R → S1
is a continuous homomorphism, then there exists c ∈ R such
that ψ(x) = eicx
for all x ∈ R.
Proof. We claim that there is a unique continuous homomorphism l : R → R such that
ψ(x) = ei·l(x)
. The exponential map ε : R → S1
given by ε(x) = eix
maps the real line around
the unit circle with period 2π and it is a universal cover. For any continuous ψ : R → S1
such that ψ(0) = 1, there exists a unique continuous lift l of this function to the real line
such that l(0) = 0, In other words, there exists a unique continuous function l : R → R such
that l(0) = 0 and ψ(x) = ε(l(x)) for all x, so the following diagram commutes:
Math 597 Fan Huang
4. A NOTE ON CHARACTER THEORY 4
R
S1R
ε
ψ
l
We also claim that if ψ is a homomorphism, then its lift l is also a homomorphism and by
Lemma 2.1 l(x) = cx for some c.
Note that ψ(s + t) = ψ(s)ψ(t), thus ε(l(s + t) − l(s) − l(t)) = 1 = e0
. It follows that
l(s + t) − l(s) − l(t) = 2πn for some n ∈ Z which depends only on s, t.
Since s and t varies continuously, we find n is a constant. We set s = t = 0 to conclude
that n = 0. Thus l is a homomorphism as promised, and so l(x) = cx for some c ∈ R by
Lemma 2.2.
Proof of proposition 2.1. Going back to the irreducible unitary representations of S1
, given
a representation ρ : S1
→ C∗
, it has a compact and thus bounded image. It follows that the
image lies on S1
. Thus ρ : S1
→ S1
is a continuous homomorphism. Precompose ρ with the
exponential map ε, then we have the following diagram commutes:
S1
S1
R
ρ
ε
ρε
By Lemma 2.2 there exists c ∈ R with ρε(x) = eicx
for all x. Thus we have ρ(eix
) = eicx
.
Note that 1 = ρ(1) = ρ(e2πi
) = ρε(2π) = e2πic
and thus c ∈ Z. So we have ρ(z) = ρn(z) =
zn
.
Finally, to show for n = k, that ρn and ρk are not isomorphic, because let f : C → C
be the G-linear map. Note that zn
f(v) = f(zn
v) = zk
f(v) for all z ∈ S1
, then we have
zn−k
= 1, i.e., n = k.
Because of the isomorphism from S1
to R/2πZ, another way to view the characters on S1
is as a map R/2πZ → C. Then the characters are given by en(x) = einx
with n ∈ Z. We
shall show all characters have this form by following an exercise from [ST03].
First we denote by ˆG the class of all characters of G, which is called the dual group of G.
We observe that the trivial character e(g) = 1 plays the role of the unit, so ˆG inherits the
structure of an abelian group under the multiplication
χ1χ2(g) = χ1(g)χ2(g)
for all g ∈ G and characters χ1, χ2.
For x ∈ G the mapping χ → χ(x) is an irreducible character of ˆG and so an element of
the dual
ˆˆG of the dual
ˆˆG of ˆG. Let φx be the element be the map that maps x ∈ ˆG to
χx. Then consider the map Φ : G →
ˆˆG given by x → φx. We see it is a homomorphism
since Φ(xy) = φxy(χ) = φx(χ)φy(χ) = Φ(x)Φ(y). Also if for x, x ∈ G, Φ(x) = Φ(x ), then
Math 597 Fan Huang
5. FAN HUANG 5
χ(x) = χ(x ) and thus χ(xx −1
) = 1 for χ ∈ ˆG, which implies xx −1
= 1. So Φ is injective.
Since G and
ˆˆG have same orders, we conclude that it is an isomophism.
Lemma 2.3. If F : R → R is continuous and F(x+y) = F(x)F(y), then F is differentiable
and F is of the form eAx
for some constant A.
Proof. If F(x) = 0 is trivial, then we have F(0) = F(0)2
. It follows that F(0) = 1 and it is
clearly differentiable and is of the form e0x
for constant 0.
Consider F(0) = 0, then for an appropriate δ, c =
δ
0
F(y) dy = 0. cF(x) =
δ
0
F(y)F(x) dy =
δ
0
F(x + y) =
δ+x
x
F(y)dy. Then
c lim
t→0
F(x + t) − F(x)
t
= lim
t→0
δ+x+t
x+t
F(y) dy −
δ+x
x
F(y) dy
t
= lim
t→0
δ+x+t
δ+x
F(y) dy
t
− lim
t→0
x+t
x
F(y) dy
t
= F(δ + x) − F(x)
Thus F is differentiable and we have
cF (x) = F(δ + x) − F(x)
= F(δ)F(x) − F(x)
F (x) =
F(δ) − 1
c
F(x)
So F is of the form eAx
for some constant A.
By applying Lemma 2.3 and noting that the characters on S1
satisfy the assumptions, we
see that all the characters are of the form eAx
for constant A . The dual group of the circle
S1
is ˆS1 = {en}n∈Z, where en(x) = einx
. Moreover, en → n gives an isomorphism between
ˆS1 and the integers Z.
Now we have a complete list of irreducible unitary representations of S1
. From [Serre77],
we already know the orthogonality relations of irreducible characters of finite groups.
(a) If χ is the character of an irreducible representation, χ, χ = 1 (i.e., χ is “of norm 1”).
(b) If χ and χ are the characters of two nonisomorphic irreducible representations, χ, χ =
0 (i.e., χ and χ are orthogonal).
(c) Let V be a linear representation of G, with character φ, and suppose V decomposes into
a direct sum of irreducible representations: V = k
i=1 Wi. Then, if W is an irreducible
representation with character χ, the number of Wi isomorphic to W is equal to the scalar
product φ, χ . Moreover, for f ∈ Cl(G), f = k f, χ χ.
As we shall see, the basic theorems from Fourier analysis give the analogy of representation
theory listed above for G = S1
and V = L2
(S1
). The space L2
(S1
) is the Hilbert space of
complex measurable functions f on S1
with
f = f 2=
1
0
|f|2
1
2
≤ ∞.
Math 597 Fan Huang
6. A NOTE ON CHARACTER THEORY 6
Also L2
(S1
) is endowed with an inner product f, g = 1
2π t∈G
f(t)g(t) dt for f, g ∈ L2
(S1
).
Note that the linear combinations of these characters f(x) = n cneinx
(x ∈ R) are trigono-
metric polynomials. With complex coefficients an trigonometric polynomial is actually a
Fourier series. Then naturally, we can find analogy with fourier analysis.
Attention is now focused on the space L2
(S1
). By Stone-Weierstrass approximation theo-
rem, the space spanned by the positive and negative powers of e2πix
is dense in the space of
continuous functions and the space C∞
(S1
) of infinitely differentiable functions is dense in
L2
(S1
). Note S1
acts by left multiplication on f ∈ L2
(S1
), i.e., (ρ(t) · f)(θ) = f(t + θ) for
t, θ ∈ S1
. This is well-defined on L2
since Lebesgue measure is invariant under translation.
Put another way, this action is indeed the regular representation and also the representation
induced from the identity element, denoted as IndG
{1}. We will talk more of the induced
representation in the next example.
From [DM72],the basic theorems of Fourier series on S1
are:
Theorem 2.1. The orthogonal family of unit length ( en = 1)
en(x) = e2πinx
, n ∈ Z
is a basis for L2
(S1
), that is, any function f ∈ L2
(S1
) can be expanded into a Fourier Series
f =
∞
−∞
ˆf(n)en
with coefficients
ˆf(n) = (f, en) = fen dx = f(x)e−2πinx
dx,
the sum being understood in the sense of distance in L2
(S1
). The map f → ˆf is therefore
an isomorphism of L2
(S1
) onto L2
(Z) and there is a Plancherel identity:
f 2
2=
1
0
|f|2
= f 2
=
∞
−∞
| ˆf(n)|2
.
Theorem 2.2. For any 1 ≤ p < ∞ and any f ∈ Cp
(S1
), the partial sums
Sn = Sn(f) =
|k|≤n
ˆf(k)ek
converge to f, uniformly as n ↑ ∞; in fact, Sn − f ∞ is bounded by a constant multiple of
n−p+1
2 .
Theorem 2.3. For functions f of class C(S1
), the arithmetic means n−1
(S0 + · · · + Sn−1)
of the partial sums Sn = |k|≤n
ˆf(k)ek converge uniformly to f.
In our example, we replace the form “ 1
|G| t∈G f(t)” with “ 1
2π
2π
0
f(t) dt ”. So the ana-
logues for the above in S1
are:
(a) The characters is of unit lenth en, en = 1
2π
2π
0
eint
e−int
dt = 1
2π
2π
0
1 dt = 1.
(b) For m = k, en, ek = 1
2π
2π
0
eint
e−ikt
dt = 1
2π
2π
0
ei(n−k)t
dt = 0.
Math 597 Fan Huang
7. FAN HUANG 7
(c) For f ∈ L2
(S1
), f = ∞
−∞ f, en en, converges for fixed n, where f, en = 1
2π
2π
0
e−int
f(t) dt,
leading to the Fourier inversion formula.
Among others, we see that the characters ρn = einx
form a complete orthonomal basis of
L2
(S1
).
3. An example: the symmetric group Σ3
(the simplest nonabelian group)
The symmetric group Σ3 is isomorphic to the dihedral group, D3, and has 6 elements (the
identity 1, three transpositions of order two: (12), (13) and (23) and two permutations of
order three: (123) and (132). It can be presented as x, y | x3
, y2
, xyxy , where x = (123)
and y = (12). Its regular representation has dimension 3! = 6 and since it is a nonabelian
finite group, its subrepresentation cannot be all 1-dimensional. In Σn, conjugacy classes
are determined by cycle decomposition sizes: two permutations are conjugate if and only
if they have the same number of cycles of each length. For Σ3, there are 3 conjugacy
classes, so there are 3 different irreducible representations over C. In the decomposition
into irreducible representations with respective dimensions d1, d2, d3, the order of the group
|Σ3|= 6 = d2
1 + d2
2 + d2
3. Thus there must be two 1-dimensional and one 2-dimensional
representation . Explicitly, they are:
(a) The trivial representation with character χT , that take the value 1 identically.
(b) The sign representation with character χΣ given by πsgn(g)v = sgn(g)v, where sgn(g) =
±1 is the sign of the permutation.
(c) The standard representation with character χA, that leaves invariant the orthogonal
subspace V = {(z1, z2, z3) ∈ C3
| z1 + z2 + z3 = 0}. This representation is 2-dimensional.
The irreducible characters of a group can be assembled into a character table, where the
first row in a character table lists representatives of conjugacy classes, the second row the
numbers of elements in the conjugacy classes, and the other rows list the values of the
characters on the conjugacy classes–which are obtained by explicitly computing traces in the
irreducible representations. The characters are constant on conjugacy classes.
The character table of Σ3 is:
Σ3 (1) (123) (12)
# 1 2 3
χT 1 1 1
χΣ 1 1 -1
χA 2 -1 0
Let H be a subgroup of G. Let ρH be the restriction of ρ, the representation of G, to H.
Let W be a sub representation of ρH, that is, a vector space of V -stable under the ρt, t ∈ H.
Denote by θ : H → GL(W) the representation of H in W thus defined. We call ρ of G in V
is induced by the representation θ of H in W and write
V ∼=
σ∈G/H
σWσ
where σW = gσW and gσ is any representative of the coset σ. Denote V =IndG
H(Wσ). Sup-
pose (V, ρ) is induced by (W, θ) and let χρ and χθ be the corresponding characters of G and
Math 597 Fan Huang
8. A NOTE ON CHARACTER THEORY 8
of H. (W, θ) determines (V, ρ) up to isomorphism.
Example 1. Let H be the trivial subgroup {1}. Let W = C be the trivial representation.
Then G/H ∼= G and thus V = g∈G gC. G acts by permuting these copies of W and we
can see that V is the regular representation of G.
As is shown in [Serre77]:
Theorem 3.1. Let h be the order of H and let R be a system of representatives of G/H.
For each u ∈ G, we have
χρ(u) =
r∈R
r−1ur∈H
χθ(r−1
ur) =
1
h s∈G
s−1us∈H
χθ(s−1
us).
In particular, χρ(u) is a linear combination of the values of χθ on the intersection of H with
the conjugacy class of u in G.
We shall apply above theorem to calculate the IndΣ3
H for the subgroups of Σ3. We note
that characters do play an important role in the algorithm. Let {ρi}r
i=1 be the irreducible
representations of G and {χi}r
i=1 be the corresponding characters. To find how the induced
representation decomposes in terms of irreducible representations of the group Σ3, that
is, IndG
Hχρ(u) = i=r
i=1 aiχi, Since the characters follow the same relation and thus ai =
χρ, χi = 1
|G| t∈G χρ(t)χi(t).
Example 2. Recall that the cyclic group Cn, a finite abelian group, has n irreducible rep-
resentations of degree 1 whoses characters are primitive roots of unity. Let H be the cyclic
subgroup C3 = {1, (123), (132)}, which is a normal subgroup of Σ3. Thus there is short exact
sequence of Σ3:
1 H = C3 Σ3
ϕ
G/H = Z2
ψ
1
The character table of C3 with 3 1-dimensional irreducible characters is
C3 (1) (123) (132)
# 1 1 1
χ1 1 1 1
χ2 1 ω ω2
χ3 1 ω2
ω
where ω = e2πi/3
.
In this example we choose a system representatives of G/H: R = {(1), (12)}. Since
H G, we have r−1
ur ∈ H for all r ∈ R and u ∈ G.
Let W1 be the 1-dimensional representation of H given by (123) → 1, i.e, the trivial
representation of C3. Then
χρ(1) = χθ(1) + χθ(1) = 2
χρ(12) = χρ(13) = χρ(23) = 0
χρ(123) = χθ(123) + χθ(132) = 2
χρ(132) = χθ(132) + χθ(123) = 2
Math 597 Fan Huang
9. FAN HUANG 9
Since IndΣ3
C3
χ1 is of the form aχT + bχΣ + cχA and we can calculate
a = χρ1 , χT =
1
6
· [2 · 1 + 2 · 1 + 2 · 1 + 0 + 0 + 0] = 1
b = χρ1 , χΣ =
1
6
· [2 · 1 + 2 · 1 + 2 · 1 + 0 + 0 + 0] = 1
c = χρ1 , χA =
1
6
· [2 · 2 + 2 · (−1) + 2 · (−1) + 0 + 0 + 0] = 0
So IndΣ3
C3
ρ1
∼= ρT ⊕ ρΣ.
Let W2 be the 1-dimensional representation of H given by (123) → ω. Then
χρ(1) = χθ(1) + χθ(1) = 2
χρ(12) = χρ(13) = χρ(23) = 0
χρ(123) = χθ(123) + χθ(132) = ω + ω2
= −1
χρ(132) = χθ(132) + χθ(123) = ω + ω2
= −1
Since IndΣ3
C3
χ2 is of the form aχT + bχΣ + cχA and we can calculate
a = χρ2 , χT =
1
6
· [2 · 1 + (−1) · 1 + (−1) · 1 + 0 + 0 + 0] = 0
b = χρ2 , χΣ =
1
6
· [2 · 1 + (−1) · 1 + (−1) · 1 + 0 + 0 + 0] = 0
c = χρ2 , χA =
1
6
· [2 · 2 + (−1) · (−1) + (−1) · (−1) + 0 + 0 + 0] = 1
So IndΣ3
C3
ρ2
∼= ρA, and same for IndΣ3
C3
ρ3 by symmetry.
Example 3. Let H be the cyclic subgroup C2 = {1, (12)} and its character table is:
C2 (1) (12)
# 1 1
χ1 1 1
χ2 1 -1
The three cosets are {(1), (12)}, {(13), (123)}, {(23), (132)}. We choose a system represen-
tatives of G/H: R = {(1), (123), (132)}. A bit different from previous example,H is not a
normal subgroup in Σ3. We need to check if r−1
ur ∈ H for all r ∈ R and u ∈ G.
For example, if u = (12), then
r ∈ R r−1
ur in H?
1 (12) Yes
(123) (13) No
(132) (23) No
if u = (123), then
r ∈ R r−1
ur in H?
1 (123) No
(123) (123) No
(132) (123) No
Math 597 Fan Huang
10. A NOTE ON CHARACTER THEORY 10
Let W1 be the 1-dimensional representation of H given by (12) → 1, i.e, the trivial repre-
sentation of C2. Then
χρ(1) = χθ(1) + χθ(1) + χθ(1) = 3
χρ(12) = χρ(13) = χρ(23) = χθ(12) = 1
χρ(123) = χρ(132) = 0
Since IndΣ3
C2
χ1 is of the form aχT + bχΣ + cχA and we can calculate
a = χρ1 , χT =
1
6
· [3 · 1 + 0 + 0 + 1 · 1 + 1 · 1 + 1 · 1] = 1
b = χρ1 , χΣ =
1
6
· [3 · 1 + 0 + 0 + 1 · (−1) + 1 · (−1) + 1 · (−1)] = 0
c = χρ1 , χA =
1
6
· [3 · 2 + 0 + 0 + 0 + 0 + 0] = 1
So IndΣ3
C2
ρ1
∼= ρT ⊕ ρA.
Let W2 be the 1-dimensional representation of H given by (12) → −1. Then
χρ(1) = χθ(1) + χθ(1) + χθ(1) = 3
χρ(12) = χρ(13) = χρ(23) = χθ(12) = −1
χρ(123) = χρ(132) = 0
Since IndΣ3
C2
χ1 is of the form aχT + bχΣ + cχA and we can calculate
a = χρ2 , χT =
1
6
· [3 · 1 + 0 + 0 + (−1) · 1 + (−1) · 1 + (−1) · 1] = 0
b = χρ2 , χΣ =
1
6
· [3 · 1 + 0 + 0 + (−1) · (−1) + (−1) · (−1) + (−1) · (−1)] = 1
c = χρ2 , χA =
1
6
· [3 · 2 + 0 + 0 + 0 + 0 + 0] = 1
So IndΣ3
C2
ρ2
∼= ρΣ ⊕ ρA.
Recall that Σ3 acts on the cosets of G/H by ggσH = gτ H and we can actually compute the
explicit induced representation. For each g ∈ G, first note if it flips cosets to decide whether
the matrix is diagonal or flipped. Then note which element of H appears and decides the
actual entries in the matrix. Specifically, the matrix representation induced from W2 in
Example 2 are as follows:
1 →
1 0
0 1
, (12) →
0 1
1 0
, (13) →
0 w
w2
0
,
(23) →
0 w2
w 0
, (123) →
w 0
0 w2 , (132) →
w2
0
0 w
Again, it is clear that this induced representation is ρA by directly calculating the trace for
each matrix .
We see from the above examples that normal subgroups give an easier way to compute
induced characters.
Another way to look at induced representation is by seeing that W is a left C[H]-module
and C[G] is a right C[H]-module. We define V := C[G] ⊗C[H] W. In fact C[G] is a free
Math 597 Fan Huang
11. FAN HUANG 11
C[H]-module via multiplication on the right with basis of gσ, representatives of left cosets
of H. Here, for example,
(123) · ((1) ⊗ e) = (123) ⊗ e = (1)((123) ⊗ e) = (1) ⊗ (123) · e.
(123) · ((12) ⊗ e) = (123)(12) ⊗ e = (13) ⊗ e = (12)(132) ⊗ e
These results lead to further questions. For example, why do characters decide the rep-
resentation up to isomorphism? Another question is how to recover the symmetric function
from the trace?
Acknowledgement
The author would like to thank Professor Matthew Ando for precious and generous guid-
ance, also appreciate undergraduate student Yan Zhou for helpful suggestions and graduate
student Sarah Loeb for careful peer review.
References
[Serre77] Jean-Pierre Serre, Linear Representations of Finite Groups, Springer-Verlag, NY,1977.
[DM72] H.Dym and H.P.Mckean, Fourier Series and Integrals, Academic Press, NY, 1972.
[ST03] Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis Fourier Analysis: An Introduc-
tion, Princeton University Press, NJ, 2003, 231–237.
[MA10] Michael Artin, Algebra, Pearson,2 edition, 2010.
[JM74] James R.Munkres, Topology; A First Course, Prentice Hall College Div, 1974.
[CT05] Constantin Teleman, Lecture Notes of Representation Theory, CT, Lent 2005, available at math.
berkeley.edu/~teleman/math/RepThry.pdf.
Department of Mathematics, University of Illinois, Urbana-Champaign, IL 61801, USA
E-mail address: fhuang5@illinois.edu
Math 597 Fan Huang