88 research outputs found
Vertices of Specht modules and blocks of the symmetric group
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
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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