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).



Center of an irreducible representation over $\mathbb{Q}$
Theorem (74.5) in Curtis' and Reiner's Methods of Representation Theory, Volume II should answer all your questions. Indeed, for an irreducible representation the center of the endomorphism algebra is the character field, i.e. the number field generated by the character values.


