Symmetric Group Character Degrees and Hook Numbers

Details Symmetric Group Character Degrees and Hook Numbers Symmetric Group Character Degrees and Hook Numbers

2007-09-06

arxiv.org/abs/0709.0897v1

Mathematics

Representation Theory

24 pages, to appear in Proc. London Math. Soc

Scientific paper

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.

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