Identification of the irreducible modular representations of \(GL_n(q)\).

*(English)*Zbl 0622.20032The 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 |

##### Keywords:

finite general linear groups; \(p\)-regular classes; \(p\)-modular irreducible representations; \((n,p)\)-indices; \(p\)-modular decomposition matrices
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\textit{R. Dipper} and \textit{G. James}, J. Algebra 104, 266--288 (1986; Zbl 0622.20032)

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##### References:

[1] | Dipper, R, On the decomposition numbers of the finite general linear groups, Trans. amer. math. soc., 290, 315-344, (1985) · Zbl 0576.20005 |

[2] | Dipper, R, On the decomposition numbers of the finite general linear groups II, Trans. amer. mah. soc., 292, 123-133, (1985) · Zbl 0579.20008 |

[3] | Dipper, R; James, G.D, Representations of Hecke algebras of general linear groups, (), 20-52 · Zbl 0587.20007 |

[4] | \scR. Dipper and G. D. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc., in press. · Zbl 0615.20009 |

[5] | Dornhoff, L, Group representation theory, part B, (1972), Dekker New York · Zbl 0236.20004 |

[6] | Fong, P; Srinivasan, B, The blocks of finite general linear and unitary groups, Invent. math., 69, 109-153, (1982) · Zbl 0507.20007 |

[7] | Green, J.A, The characters of the finite general linear groups, Trans. amer. math. soc., 80, 402-447, (1955) · Zbl 0068.25605 |

[8] | James, G.D, The representation theory of the symmetric groups, () · Zbl 0393.20009 |

[9] | James, G.D, The irreducible representations of the finite general linear groups, (), 236-268 · Zbl 0587.20022 |

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