The Schur indices of the irreducible characters of \(G_ 2(2^ n)\) (Q1327783)

Let \(G_2(q)\) be the finite Chevalley group of type \((G_2)\) over a finite field \(\mathbb{F}_q\) with \(q\) elements. It was shown by the second author [Tokyo J. Math. 8, 133-150 (1985; Zbl 0584.20032)] that the following theorem holds for odd \(q\): Theorem. The Schur index \(m_\mathbb{Q}(\chi)\) of any complex irreducible character \(\chi\) of \(G_2(q)\) with respect to \(\mathbb{Q}\) is equal to 1. In this paper we prove that the theorem holds also for \(q=2^n\).