88 research outputs found

    Vertices of Specht modules and blocks of the symmetric group

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    This paper studies the vertices, in the sense defined by J. A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. We also use it to find the Green correspondents of a particular family of simple Specht modules; as a corollary, we get a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, as developed by M. Brou{\'e}, together with combinatorial arguments using Young tableaux.Comment: 18 pages, 1 figur

    Set families and Foulkes modules

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    We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label irreducible characters appearing as summands of the characters of Foulkes modules. The homomorphisms are defined using certain families of subsets of the natural numbers. These families are of independent interest; we prove a number of combinatorial results concerning them.Comment: 22 pages, 3 figures, final published versio
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