Defect of characters of the symmetric group

Following the work of B. Kuelshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the \ell-defect of characters of the symmetric group S_n, where \ell > 1 is an arbitrary integer. We prove that the \ell -defect is given by an analogue of the hook-length formula, and use it to prove, when n < \ell^2, an \ell-version of the McKay Conjecture in S_n .

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