Branching Rules for Modular Representations of Symmetric Groups Iii: Some Corollaries and a Problem of Mullineux
نویسنده
چکیده
Let K be a field of characteristic p > 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first« — 1 letters, and D k the irreducible ATn-module corresponding to a (/^-regular) partition X of n. In [9] we described the socle of the restriction D [T and obtained a number of other results which can be considered as generalisations of the classical Branching Theorem in zero characteristic (cf. for example [4, 9.2]). The purpose of this paper is to provide some applications of these results. First we obtain a lower bound for the dimension of an irreducible module D in terms of the paths in a modular version of the Young graph and (which is essentially the same) in terms of a modular version of the standard A-tableaux. Then we prove a somewhat surprising result, namely that soc (D | L ) contains at most p simple components (it was shown in [7] that this socle is multiplicity-free). Moreover we find explicitly the number of these simple components and prove that they all belong to distinct blocks. We also obtain some information about the indecomposable components of D I Let
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