9Dual spaces and tensor products of representations
II Representation Theory
9.4 Characters of G × H
We have looked at characters of direct products a bit before, when we decomposed
an abelian group into a product of cyclic groups. We will now consider this in
the general case.
Proposition.
Let
G
and
H
be two finite groups with irreducible characters
χ
1
, ··· , χ
k
and
ψ
1
, ··· , ψ
r
respectively. Then the irreducible characters of the
direct product G × H are precisely
{χ
i
ψ
j
: 1 ≤ i ≤ k, 1 ≤ j ≤ r},
where
(χ
i
ψ
j
)(g, h) = χ
i
(g)ψ
j
(h).
Proof.
Take
ρ
:
G → GL
(
V
) affording
χ
, and
ρ
0
:
H → GL
(
W
) affording
ψ
.
Then define
ρ ⊗ρ
0
: G ×H → GL(V ⊗ W )
(g, h) 7→ ρ(g) ⊗ρ
0
(h),
where
(ρ(g) ⊗ ρ
0
(h))(v
i
⊗ w
j
) 7→ ρ(g)v
i
⊗ ρ
0
(h)w
j
.
This is a representation of
G × H
on
V ⊗ W
, and
χ
ρ⊗ρ
0
=
χψ
. The proof is
similar to the case where
ρ, ρ
0
are both representations of
G
, and we will not
repeat it here.
Now we need to show
χ
i
ψ
j
are distinct and irreducible. It suffices to show
they are orthonormal. We have
hχ
i
ψ
j
, χ
r
ψ
s
i
G×H
=
1
|G ×H|
X
(g,h)∈G×H
χ
i
ψ
j
(g, h)χ
r
ψ
s
(g, h)
=
1
|G|
X
g∈G
χ
i
(g)χ
r
(g)
1
|H|
X
h∈H
ψ
j
(h)ψ
s
(h)
!
= δ
ir
δ
js
.
So it follows that
{χ
i
ψ
j
}
are distinct and irreducible. We need to show this is
complete. We can consider
X
i,j
χ
i
ψ
j
(1)
2
=
X
χ
2
i
(1)ψ
2
j
(1) =
X
χ
2
i
(1)
X
ψ
2
j
(1)
= |G||H| = |G × H|.
So done.