A **rational character** of a finite group $G$ is complex character taking values in $\Q$. Such a character can be obtained by taking the sum of Galois conjugates of an irreducible complex character, and we will refer to these as **irreducible rational characters** even though they are not irreducible as complex characters. The irreducible rational characters form a $\Q$-basis for the $\Q$-vector space of rational characters.

Note that the representation corresponding to a rational character may not take values in $\GL_n(\Q)$ even though the character takes values in $\Q$ (the difference between the field of character values and the field in which matrix entries lie is measured by the Schur index).

The value taken by a rational character is constant on each rational conjugacy classes.

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- Review status: beta
- Last edited by David Roe on 2021-09-29 02:34:56

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