Ž . Journal of Algebra 228, 119142 2000 doi:10.1006jabr.1999.8248, available online at http:www.idealibrary.com on Decomposition Numbers of Symmetric Groups by Induction Gordon James and Adrian Williams Department of Mathematics, Imperial College of Science, Technology and Medicine, Queen’s Gate, London SW7 2BZ, United Kingdom Communicated by Michel Broue ´ Received September 1, 1999 1. INTRODUCTION The problem of determining the p-modular decomposition matrix of the symmetric group  appears to be difficult. In this paper, we shall exploit n recent results of Kleshchev in order to relate certain decomposition numbers of  to decomposition numbers of smaller symmetric groups. n   First, we recall some established results 4, 6 on the representation Ž . theory of  .A partition    ,  ,... of n is a non-increasing n 1 2 sequence of non-negative integers  whose sum is n. Let p be a prime i number and let F be a field of characteristic p. Then for each partition  of n there exists an F  -module S  which is known as a Specht module. n Ž . A partition    ,  ,... of n is p-regular if no non-zero part  1 2 i occurs p or more times, and the irreducible F  -modules D  are n indexed by the p-regular partitions  of n. Given F  -modules M and n 1     M , we write M : D for the composition multiplicity of D in M and 2 1 1       we write M  M if M : D  M : D for all the p-regular parti- 1 2 1 2 tions  of n.     The integers S : D are the p-modular decomposition numbers of      , and no method is available, at present, for evaluating S : D , in n general.     Let  be a p-regular partition of n. It is known that S : D  1 and     S : D  0 only if   , where  denotes the dominance order on         the partitions of n 4, 3.2 . In particular, S : D  0 only if the number of non-zero parts of  does not exceed the number of non-zero parts of .     Therefore, to gain insight into the decomposition numbers S : D , it is sensible to consider cases where  has few non-zero parts. The answers   are known when  has no more than two non-zero parts 2, 3 and this 119 0021-869300 $35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved. w metadata, citation and similar papers at core.ac.uk