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The characters of $$G_ 2(2^ n)$$. (English) Zbl 0614.20028
Let $$G_ 2(p^ n)$$ 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