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Identification of the irreducible modular representations of \(GL_n(q)\). (English) Zbl 0622.20032
The characters of the finite general linear group \(G=GL(n,q)\) were constructed by J. A. Green [Trans. Am. Math. Soc. 80, 402-447 (1955; Zbl 0068.25605)] who moreover constructed a bijection between the conjugacy classes and the ordinary irreducible characters of the group. In this paper the authors give a bijection between the \(p\)-regular classes of \(G\), where \(p\) is a prime not dividing \(q\), and the \(p\)-modular irreducible representations of \(G\). They define a set of arrays called \((n,p)\)-indices, and subsets of this set called head, foot and special indices. Then the head \((n,p)\)-indices parametrize the \(p\)-regular conjugacy classes of \(G\). For each \((n,p)\)-index \(I\) a \(p\)-modular representation \(D(I)\) is constructed using results G. James [in Proc. Lond. Math. Soc., III. Ser. 52, 236-268 (1986; Zbl 0587.20022)]. It is then shown that if \(I\) is a special food index then \(D(I)\) is the module constructed by R. Dipper [in Trans. Am. Math. Soc. 290, 315-344 (1985; Zbl 0576.20005)]. A bijection \(I\to I'\) is set up between the set of head \((n,p)\)-indices and the set special foot \((n,p)\)-indices. The main theorem of this paper is then that if \(I\) is an \((n,p)\)-index which is not a foot, and if \(I\to I'\) then \(D(I)\) is isomorphic to \(D(I')\). Thus if \(I\) runs over the head \((n,p)\)-indices then \(D(I)\) runs over a complete set of inequivalent irreducible \(p\)-modular representations, and this gives the bijection mentioned above. Some consequences of the main theorem concerning the \(p\)-modular decomposition matrix of \(G\) are derived in the last section of the paper.
Reviewer: B.Srinivasan

MSC:
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C20 Modular representations and characters
20C30 Representations of finite symmetric groups
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