Irreducible modules for classical and alternating groups.
From the introduction: In this article we consider the following question: Given finite groups \(H<G\) and an absolutely irreducible \(\mathbb{F} G\)-module \(M\), when does \(M\!\!\downarrow_H\) remain irreducible? The question arises naturally in the study of the maximal subgroups of the classical groups. We assume that \(H\cong A_n\) or \(2.A_n\), \(n\geq 15\) and that \(G\) is a simply connected classical group over \(\mathbb{F}\), where \(\mathbb{F}\) is an algebraically closed field of characteristic \(p\geq 0\). In particular \(G=\text{SL}(V)\), \(\text{Sp}(V)\) or \(\text{Spin}(V)\) where \(V\) be a vector space over \(\mathbb{F}\). For convenience, we assume that \(\dim(V)\) is even whenever \(G=\text{Spin}(V)\) and \(p=2\). In this context we prove the following: Theorem 1.1. Suppose that \(M=M(\lambda)\) is a tensor indecomposable \(\mathbb{F} G\)-module such that \(M\!\!\downarrow_H\) is absolutely irreducible and \(\dim(M)>\dim(V)\). 1. If \(G=\text{SL}(V)\), then either (a) \(\dim(V)\leq n^3\) or (b) \(\lambda\) or \(\rho(\lambda)=3\lambda_1\), \(2\lambda_1+\lambda_2\), \(2\lambda_1+\lambda_\ell\), \(\lambda_2+\lambda_\ell\) or \(\lambda_3\) and \(\dim(V)\leq n^4\). 2. If \(G=\text{Sp}(V)\) or \(\text{Spin}(V)\), then either (a) \(\dim(V)\leq 2n^4\) or (b) \(\lambda=3\lambda_1\), \(2\lambda_1+\lambda_2\) or \(\lambda_3\) and \(\dim(V)\leq 2n^6\). Furthermore, if \(n\geq 74\) and \(p\neq 2\), then \(G\neq\text{Sp}(V)\). -- This result strengthens the main result of the author [Pac. J. Math. 192, No. 2, 297-306 (2000; Zbl 1009.20053)].
scientific article; zbMATH DE number 1448621
scientific article