In (18.76) and (18.78), we have shown the orthogonality relationship between
traces. We have another important orthogonality relationship between them. To prove
this theorem, we need a notion of group algebra [5]. The argument is as follows:
(a) Let us think of a set comprising group elements expressed as
ℵ ¼
X
g
a g g,
ð18:101Þ
where g is a group element of a group ℊ ¼ {g 1 , g 2 , Á Á Á, g n }; a g is an arbitrarily
chosen complex number; Σ g means that summation should be taken over group
elements. Let ℵ
0 be another set similarly defined as (18.101). That is,
ℵ
0
¼
X
g 0 a
0
g 0 g
0
:
ð18:102Þ
Then we can define a following sum:
ℵ þ ℵ
0
¼
X
g
a g g þ
X
g 0 a
0
g 0 g
0
¼
X
g
a g g þ a
0
g g
¼
X
g
a g þ a
0
g
g:
ð18:103Þ
Also, we get
ℵ Á ℵ
0
¼
X
g
a g g
X
g 0 a
0
g 0 g
0
¼
X
g 0
X
g
a g gg
0
a
0
g 0
¼
X
g 0 À1
X
g
a g gg
0 À1
a
0
g 0 À1 ¼
X
g 0À1
X
gg 0 a gg 0 gg
0 g
0 À1
a
0
g 0 À1
¼
X
g 0À1
X
gg 0 a gg 0 g
a
0
g 0 À1 ¼
X
gg 0
X
g 0À1 a gg 0 a
0
g 0 À1
g ¼
X
g
X
g 0 a gg 0 a
0
g 0 À1
g,
ð18:104Þ
where we used the rearrangement theorem and suitable exchange of group
elements. Thus, we see that the above-defined set ℵ is closed under summations
(i.e., linear combinations) and multiplications. A set closed under summations
and multiplications is said to be an algebra. If the set forms a group, the said set is
called a group algebra.
So far we have treated calculations as a multiplication between two elements
g and g
0
, i.e., g ⋄ g
0 (see Sect. 16.1). Now we start regarding the calculations as
summation as well. In that case, group elements act as basis vectors in a vector
space. Bearing this in mind, let us further define specific group algebra.
(b) Let K i be the i-th conjugacy class of a group ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. Also let K i be
such that
K i ¼ A
i
ð Þ
1 , A
i
ð Þ
2 , Á Á Á, A
i
ð Þ
k i
n
o
,
ð18:105Þ
where k i is the number of elements belonging to K i . Now think of a set
gK i g
À1 (
8
g 2 ℊ). Then, due to the definition of a class, we have
704
18 Representation Theory of Groups
traces. We have another important orthogonality relationship between them. To prove
this theorem, we need a notion of group algebra [5]. The argument is as follows:
(a) Let us think of a set comprising group elements expressed as
ℵ ¼
X
g
a g g,
ð18:101Þ
where g is a group element of a group ℊ ¼ {g 1 , g 2 , Á Á Á, g n }; a g is an arbitrarily
chosen complex number; Σ g means that summation should be taken over group
elements. Let ℵ
0 be another set similarly defined as (18.101). That is,
ℵ
0
¼
X
g 0 a
0
g 0 g
0
:
ð18:102Þ
Then we can define a following sum:
ℵ þ ℵ
0
¼
X
g
a g g þ
X
g 0 a
0
g 0 g
0
¼
X
g
a g g þ a
0
g g
¼
X
g
a g þ a
0
g
g:
ð18:103Þ
Also, we get
ℵ Á ℵ
0
¼
X
g
a g g
X
g 0 a
0
g 0 g
0
¼
X
g 0
X
g
a g gg
0
a
0
g 0
¼
X
g 0 À1
X
g
a g gg
0 À1
a
0
g 0 À1 ¼
X
g 0À1
X
gg 0 a gg 0 gg
0 g
0 À1
a
0
g 0 À1
¼
X
g 0À1
X
gg 0 a gg 0 g
a
0
g 0 À1 ¼
X
gg 0
X
g 0À1 a gg 0 a
0
g 0 À1
g ¼
X
g
X
g 0 a gg 0 a
0
g 0 À1
g,
ð18:104Þ
where we used the rearrangement theorem and suitable exchange of group
elements. Thus, we see that the above-defined set ℵ is closed under summations
(i.e., linear combinations) and multiplications. A set closed under summations
and multiplications is said to be an algebra. If the set forms a group, the said set is
called a group algebra.
So far we have treated calculations as a multiplication between two elements
g and g
0
, i.e., g ⋄ g
0 (see Sect. 16.1). Now we start regarding the calculations as
summation as well. In that case, group elements act as basis vectors in a vector
space. Bearing this in mind, let us further define specific group algebra.
(b) Let K i be the i-th conjugacy class of a group ℊ ¼ {g 1 , g 2 , Á Á Á, g n }. Also let K i be
such that
K i ¼ A
i
ð Þ
1 , A
i
ð Þ
2 , Á Á Á, A
i
ð Þ
k i
n
o
,
ð18:105Þ
where k i is the number of elements belonging to K i . Now think of a set
gK i g
À1 (
8
g 2 ℊ). Then, due to the definition of a class, we have
704
18 Representation Theory of Groups