On quotients of Hom-functors and representations of finite general linear groups. II
This is the second article on quotients of Hom-functors and their applications to the representation theory of finite general linear groups in a nondescribing characteristic [for part I see ibid. 130, No. 1, 235-259 (1990; Zbl 0698.20005)]. After some general results on quotients of Hom-functors and their connection to the Harish-Chandra theory these constructions are used to obtain a full classification of the \(\ell\)-modular irreducible representations of \(\text{GL}_n(q)\) for some prime power \(q\) which is not divisible by the prime \(\ell\) and to explain some facts on their Harish-Chandra series and decomposition numbers. The paper is organized as follows: In the first three sections the general theory of those functors is further developed. The relation between the functors for different projective resolutions is investigated as well as the image of the different right inverses of those. The second part of this article, starting with Section 4, is devoted to the representation theory of \(\text{GL}_n(q)\). As a consequence he gets a classification of the irreducible \(\ell\)-modular representations of \(G=\text{GL}_n(q)\) for primes \(\ell\) not dividing \(q\) and gets information on decomposition numbers in terms of Hecke- and \(q\)-Schur algebras. At the end of Section 6 he recovers and generalizes a result of Takeuchi.
scientific article; zbMATH DE number 1234248
scientific article