Modular Representations of Symmetric Groups

D. E. Littlewood

Abstract

An analysis of partitions is made by considering the congruence of sets of numbers, which is to some extent equivalent to, but more far-reaching than Nakayama's method of hooks. A proof is given of Nakayama's conjecture concerning the p-blocks of symmetric group characters which is much simpler than the recent proof by Brauer and Robinson. The p-residue and p-quotient of a partition are defined, and a formula is obtained relating to symmetric group characteristics. A procedure is described whereby the mode of separation in every case may be determined, of the 0-characters of the symmetric groups into p-characters. The method of p-residues and p-quotients is employed to give a method of expansion for the plethysm of S-functions which is unambiguous and yet reasonably concise.