For a finite group \(G\), what is its complex character table? In what fields do its entries lie?
Solution
Choose representatives \[ g_1,\ldots,g_r \] of the conjugacy classes of \(G\), and choose the irreducible complex characters \[ \chi_1,\ldots,\chi_r. \] The character table is the matrix \[ (\chi_i(g_j))_{i,j}. \] Rows are indexed by irreducible complex representations and columns by conjugacy classes.
If \(g\in G\) has order \(m\), then in any finite-dimensional complex representation the eigenvalues of \(\rho(g)\) are \(m\)-th roots of unity. Hence \[ \chi(g)=\operatorname{tr}(\rho(g)) \] is a sum of \(m\)-th roots of unity. Therefore every character value is an algebraic integer lying in a cyclotomic field.
If \(e\) is the exponent of \(G\), then every element order divides \(e\), so the entire character table has entries in \[ \QQ(\zeta_e). \] Since \(e\mid |G|\) for finite groups, one may also use the larger field \[ \QQ(\zeta_{|G|}). \] The entries need not be rational or real in general.