# Center of an irreducible representation over $\mathbb{Q}$

Let $G$ be a finite group, $\rho\colon G \rightarrow \mathrm{GL}_n(\mathbb{Q})$ its irreducible representation, and $D$ the division algebra of $G$-endomorphisms of $\mathbb{Q}^n$. The division $\mathbb{Q}$-algebra $D$ is finite dimensional. What is its center?

I've heard that the center is the number field generated by the values of the character of an irreducible constituent of $\rho_{\mathbb{C}}$ but I cannot see this (and I don't know where to look this up).

• Here is a hint: the center $Z$ of $\mathbb{Q}[G]$ is the subalgebra of functions $G \rightarrow \mathbb{Q}$ constant on the conjugacy class. Because they are central, each element of $Z$ acts on a representation as an endomorphism. Finally, On a complex irreducible representation each element of $Z$ acts has an endomorphism, hence as a complex number by schur lemma... But then the number in question has to be the trace/dimension hence easily relates to the value of the character. – Simon Henry Aug 20 '15 at 17:41
• the question mathoverflow.net/questions/119658/… is related to this question – Venkataramana Oct 2 '15 at 12:25