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Proceedings of the London Mathematical Society Advance Access originally published online on August 11, 2007
Proceedings of the London Mathematical Society 2008 96(1):26-50; doi:10.1112/plms/pdm028
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© 2007 London Mathematical Society

Symmetric group character degrees and hook numbers

David A. Craven

Mathematical Institute
University of Oxford
24–29 St Giles’
Oxford
OX1 3LB
United Kingdom

Received 21 July 2006.

In this article we prove the following result: for any two natural numbers k and {ell}, and for all sufficiently large symmetric groups Sn, there are k disjoint sets of {ell} irreducible characters of Sn, 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 Moretó in [5]. 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.

2000 Mathematics Subject Classification 20C30.


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