# Irreducible representation

• Dec 1st 2010, 01:56 AM
bram kierkels
Irreducible representation
Let $\displaystyle G$ be a group and let $\displaystyle z$ be in the center of $\displaystyle G$.
Why is $\displaystyle \rho(z)$ for every irreducible representation $\displaystyle \rho$ of $\displaystyle G$ a scalair?
• Dec 1st 2010, 01:00 PM
Drexel28
Quote:

Originally Posted by bram kierkels
Let $\displaystyle G$ be a group and let $\displaystyle z$ be in the center of $\displaystyle G$.
Why is $\displaystyle \rho(z)$ for every irreducible representation $\displaystyle \rho$ of $\displaystyle G$ a scalair?

What do you define as rho?
• Dec 2nd 2010, 02:51 AM
bram kierkels
Quote:

Originally Posted by Drexel28
What do you define as rho?

rho is just an arbitrary representation, thus $\displaystyle \rho:G\rightarrow GL(V)$ is a homomorphism on a (complex) vector space $\displaystyle V$, where GL(V) is the general lineair group.
I think it is easier then it looks:
Since $\displaystyle \rho(z):V\rightarrow V$, and $\displaystyle \rho(z)\rho(x)=\rho(zx)=\rho(xz)=\rho(x)\rho(z).$, it follows directly from this corollary of Schur's Lemma:
PlanetMath: Schur's lemma