Download PDF

character∗
djao†
2013-03-21 13:25:44
Let ρ : G −→ GL(V ) be a finite dimensional representation of a group G
(i.e., V is a finite dimensional vector space over its scalar field K). The character
of ρ is the function χV : G −→ K defined by
χV (g) := Tr(ρ(g))
where Tr is the trace function.
Properties:
• χV (g) = χV (h) if g is conjugate to h in G. (Equivalently, a character is a
class function on G.)
• If G is finite, the characters of the irreducible representations of G over
the complex numbers form a basis of the vector space of all class functions
on G (with pointwise addition and scalar multiplication).
• Over the complex numbers, the characters of the irreducible representations of G are orthonormal under the inner product
(χ1 , χ2 ) :=
1 X
χ1 (g)χ2 (g)
|G|
g∈G
∗ hCharacteri
created: h2013-03-21i by: hdjaoi version: h31843i Privacy setting: h1i
hDefinitioni h20C99i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1