The rational representation ring $R_{\mathbb{Q}}(G)$ and rational-valued class functions

Let $G$ be a finite group, and let $R(G)$ be its complex character ring (Virtual characters and the character ring $R(G)$ of a finite group).

The rational representation ring $R_{\mathbb{Q}}(G)$ is the subgroup of $R(G)$ generated by the complex characters of finite-dimensional $\mathbb{Q}$-representations of $G$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). Equivalently, $R_{\mathbb{Q}}(G)$ is the Grothendieck group of finite-dimensional $\mathbb{Q}G$-modules, viewed inside the complex character ring by extension of scalars from $\mathbb{Q}$ to $\mathbb{C}$.

A class function $\alpha :G\to \mathbb{C}$ is rational-valued when $\alpha (g)\in \mathbb{Q}$ for every $g\in G$.

• On this page, an unqualified rational character means an element of $R_{\mathbb{Q}}(G)$, not merely a rational-valued class function.

• Every element of $R_{\mathbb{Q}}(G)$ is rational-valued, because the trace of a $\mathbb{Q}$-linear operator is rational and the character-ring operations are integral linear combinations of such traces.

• The converse can fail: the companion page records a rational-valued irreducible character of $Q_8$ that is not afforded by any $\mathbb{Q}$-representation.