The characters of a representation D of G is: X(g)=Tr(D(g))=trace of the D(g)'s

Theorem:

* a,b in G with a~b => X(a)=X(b) (their characters are the same)

Proof:

D(b)=D(g^ag)=D(g^)D(a)D(g)

X(b)=Tr(D(b))=Tr(D(g^)D(a)D(g))

=> Tr(D(b))=Tr(D(a))

- two equivalent representations of G have the same character

Proof as above

- The character of reducible representations is the sum ov the characters of the component representations

X(g) = X1(g)+X2(g)

- Character of e = dimension of group.
- a finite group G has k inequivalent irreducible representations D1, D2,..,Dk and l conjugacy classes C1,C2,...,Cn with k=l.

source

jl@crush.caltech.edu index

young_diagram

irreducible

pseudoreal

abelian

weyl

SO

GL