9Dual spaces and tensor products of representations
II Representation Theory
9.7 Character ring
Recall that
C
(
G
) is a commutative ring, and contains all the characters. Also,
the sum and products of characters are characters. However, they don’t quite
form a subring, since we are not allowed to subtract.
Definition
(Character ring)
.
The character ring of
G
is the
Z
-submodule of
C(G) spanned by the irreducible characters and is denoted R(G).
Recall we can obtain all characters by adding irreducible characters. However,
subtraction is not allowed. So the charactering ring includes something that is
not quite a character.
Definition
(Generalized/virtual characters)
.
The elements of
R
(
G
) are called
generalized or virtual characters. These are class functions of the form
ψ =
X
χ
n
χ
χ,
summing over all irreducibles χ, and n
χ
∈ Z.
We know
R
(
G
) is a ring, and any generalized character is a difference of two
characters (we let
α =
X
n
χ
≥0
n
χ
χ, β =
X
n
χ
<0
(−n
χ
)χ.
Then
ψ
=
α − β
, and
α
and
β
are characters). Then the irreducible characters
{χ
i
}
forms a
Z
-basis for
R
(
G
) as a free
Z
-module, since we have shown that
they are independent, and it generates R(G) by definition.
Lemma.
Suppose
α
is a generalized character and
hα, αi
= 1 and
α
(1)
>
0.
Then α is actually a character of an irreducible representation of G.
Proof. We list the irreducible characters as χ
1
, ··· , χ
k
. We then write
α =
X
n
i
χ
i
.
Since the χ
i
’s are orthonormal, we get
hα, αi =
X
n
2
i
= 1.
So exactly one of
n
i
is
±
1, while the others are all zero. So
α
=
±χ
i
for some
i
.
Finally, since
α
(1)
>
0 and also
χ
(1)
>
0, we must have
n
i
= +1. So
α
=
χ
i
.
Henceforth we don’t distinguish between a character and its negative, and
often study generalized characters of inner product 1 rather than irreducible
characters.