This is the first in a series of three very important papers by the author concerned with the following question: Given a \(p\)-modular irreducible representation \(D^\lambda\) of the symmetric group \(S_n\), which modular irreducible representations occur in the restriction of \(D^\lambda\) to \(S_{n-1}\) [for part II cf. J. Reine Angew. Math. 459, 163-212 (1995; 817.20009), part III, J. Lond. Math. Soc., II. Ser. 54, No. 1, 25-38 (1996; see following review Zbl 0854.20014)].
The present paper proves three conjectures of D. Benson (posed for \(p=2\)) in arbitrary characteristic. They have to do with the questions about when the restriction is irreducible, (the Jantzen-Seitz conjecture) and about the socle of the restriction. The proofs are obtained by transforming the problems into corresponding problems in the group \(\text{SL}_n(K)\), where \(K\) is an algebraically closed field of characteristic \(p\).