Chapter 18
Representation Theory of Groups
Representation theory is an important pillar of the group theory. As we shall see
soon, the word “representation” and its definition sound a bit daunting. To be short,
however, we need “numbers” or “matrices” to do a mathematical calculation.
Therefore, we may think of the representation as merely numbers and matrices.
Individual representations have their dimension. If that dimension is one, we treat a
representation as a number (real or complex). If the dimension is two or more, we are
going to deal with matrices; in the case of the n-dimension, it is a (n, n) square
matrix. In this chapter we focus on the representation theory of finite groups. In this
case, we have an important theorem stating that a representation of any finite group
can be converted to a unitary representation by a similarity transformation. That is,
group elements of a finite group are represented by a unitary matrix. According to the
dimension of the representation, we have the same number of basis vectors. Bearing
these things firmly in mind, we can pretty easily understand this important notion of
representation.
18.1 Definition of Representation
In Sect. 16.4 we dealt with various aspects of the mapping between group elements.
Of these, we studied fundamental properties of isomorphism and homomorphism. In
this section we introduce the notion of representation of groups and study it. If we
deal with a finite group consisting of n elements, we describe it as ℊ ¼ {g 1 e,
g 2 , Á Á Á, g n } as in the case of Sect. 16.1.
Definition 18.1 Let ℊ ¼ {g ν } be a group comprising elements g ν . Suppose that a
(d, d ) matrix D(g ν ) is given for each group element g ν . Suppose also that in
correspondence with
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_18
679
Representation Theory of Groups
Representation theory is an important pillar of the group theory. As we shall see
soon, the word “representation” and its definition sound a bit daunting. To be short,
however, we need “numbers” or “matrices” to do a mathematical calculation.
Therefore, we may think of the representation as merely numbers and matrices.
Individual representations have their dimension. If that dimension is one, we treat a
representation as a number (real or complex). If the dimension is two or more, we are
going to deal with matrices; in the case of the n-dimension, it is a (n, n) square
matrix. In this chapter we focus on the representation theory of finite groups. In this
case, we have an important theorem stating that a representation of any finite group
can be converted to a unitary representation by a similarity transformation. That is,
group elements of a finite group are represented by a unitary matrix. According to the
dimension of the representation, we have the same number of basis vectors. Bearing
these things firmly in mind, we can pretty easily understand this important notion of
representation.
18.1 Definition of Representation
In Sect. 16.4 we dealt with various aspects of the mapping between group elements.
Of these, we studied fundamental properties of isomorphism and homomorphism. In
this section we introduce the notion of representation of groups and study it. If we
deal with a finite group consisting of n elements, we describe it as ℊ ¼ {g 1 e,
g 2 , Á Á Á, g n } as in the case of Sect. 16.1.
Definition 18.1 Let ℊ ¼ {g ν } be a group comprising elements g ν . Suppose that a
(d, d ) matrix D(g ν ) is given for each group element g ν . Suppose also that in
correspondence with
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_18
679