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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm032, 24 pages, doi:10.1093/imrn/rnm032 published on June 22, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

An Extension of James's Conjecture

Matthew Fayers

Queen Mary, University of London, Mile End Road, London E1 4NS, U.K.

Correspondence: Correspondence to be sent to: m.fayers{at}qmul.ac.uk

Let B be a block of an Iwahori–Hecke algebra or q-Schur algebra of the symmetric group. The decomposition matrix for B may be obtained from the decomposition matrix of the corresponding block B' in infinite characteristic by post-multiplying by an adjustment matrix; since (by a deep theorem of Ariki) there is an algorithm for computing the decomposition matrix for B', the hard part of the decomposition number problem for B is to find the adjustment matrix. James's Conjecture suggests a sufficient condition for this adjustment matrix to be the identity matrix. We extend James's Conjecture to give a necessary and sufficient condition, and prove the necessity of our condition.

Communicated by Andrei Zelevinsky


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