Permutation representations and a fragment of the decomposition matrix of symplectic and special linear groups over a finite field (Q1174011)

Given a prime \(p\) and \(q=p^ d\), an explicit construction of a fragment of the decomposition matrix of the groups \(SL_ n(q)\) and \(Sp_ n(q)\) is given for any \(n\). The basic method is the analysis of the modular permutation representation over an algebraically closed field of characteristic \(p\). At the same time, composition factors of this representation with their multiplicities are described.