The characters of \(G_ 2(2^ n)\).

*(English)*Zbl 0614.20028Let \(G_ 2(p^ n)\) be the finite Chevalley group of type \(G_ 2\) defined over a finite field of characteristic p. The character table of \(G_ 2(p^ n)\) has already been obtained by B. Chang, R. Ree and H. Enomoto for the case \(p\neq 2\). In the present paper, the authors calculate all the complex irreducible characters of the group \(G_ 2(2^ n)\). Each of these irreducible characters is expressed as a linear combination of induced characters with integral coefficients. To calculate these characters, the authors first consider the irreducible characters of a Borel subgroup B of \(G=G_ 2(2^ n)\) and next they make up the character tables of two parabolic subgroups P and Q containing B. On the other hand, the group G has subgroups L and M isomorphic to \(SL_ 3(2^ n)\) and \(SU_ 3(2^{2n})\), respectively. The character tables of L and M are well known. Then the authors obtain all the irreducible characters of G by combining the characters induced from P, Q, L and M. Finally, the detailed results of this paper are summarized in tabular form at the end of this paper.

Reviewer: Shi Jianyi

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |