The modular characters of the symmetric groups

Abstract

The ordinary irreducible characters X, $ ... 3: X, ;? 0, X, + Aa + ... + A, = n. Similarly, for a given prime number p, the modular irreducible Brauer characters 4” and the modular principal indecomposable characters 7” may be labeled by the p-regular partitions p of n, that is, partitions in which no part is repeated p times or more. This is simply due to the well-known fact (see, e.g., [8, p. 1141) that the number of p-regular conjugacy classes of S, is equal to the number of p-regular partitions of IZ. The lack of a good labeling of characters in the modular theory has always contrasted strongly with the very satisfactory state of affairs in the classical ordinary theory. In this paper we introduce relatively simple combinatorial notions in order to prove the existence of “good” labeling in the modular theory. We use the p-graph of a partition (first introduced by Littlewood [5]) in order to associate with each p-regular partition p a set 1(p) of partitions of n, by means of a special process of stripping and reassembly of the nodes. Littlewood’s p-graph