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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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##### References:
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