Foulkes modules and decomposition numbers of the symmetric group

Eugenio Giannelli, Mark Wildon

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The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in characteristic zero. The main theorem of this paper gives a combinatorial description of certain columns of the decomposition matrices of symmetric groups in odd characteristic. The result applies to certain blocks of arbitrarily high weight. It is obtained by studying the p-local structure of certain twists of the permutation module given by the action of the symmetric group of even degree 2m on the collection of set partitions of a set of size 2m into m sets each of size two. In particular, the vertices of the indecomposable summands of all such modules are characterized; these summands form a new family of indecomposable p-permutation modules for the symmetric group. As a further corollary it is shown that for every natural number w there is a diagonal Cartan number in a block of the symmetric group of weight w equal to w+1.
Original languageEnglish
Pages (from-to)255-276
Number of pages22
JournalJournal of Pure and Applied Algebra
Issue number2
Early online date26 Apr 2014
Publication statusPublished - Feb 2015

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