350 research outputs found
Symmetric Group Character Degrees and Hook Numbers
In this article we prove the following result: that for any two natural
numbers k and j, and for all sufficiently large symmetric groups Sym(n), there
are k disjoint sets of j irreducible characters of Sym(n), such that each set
consists of characters with the same degree, and distinct sets have different
degrees. In particular, this resolves a conjecture most recently made by
Moret\'o. The methods employed here are based upon the duality between
irreducible characters of the symmetric groups and the partitions to which they
correspond. Consequently, the paper is combinatorial in nature.Comment: 24 pages, to appear in Proc. London Math. So
Normal Subsystems of Fusion Systems
In this article we prove that for any saturated fusion system, that the
(unique) smallest weakly normal subsystem of it on a given strongly closed
subgroup is actually normal. This has a variety of corollaries, such as the
statement that the notion of a simple fusion system is independent of whether
one uses weakly normal or normal subsystems. We also develop a theory of weakly
normal maps, consider intersections and products of weakly normal subsystems,
and the hypercentre of a fusion system
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