In this paper we present, for any integer d, a description of the set of hooks in a d-symbol. We then introduce generalized hook length functions for a d-symbol, and prove a general result about them, involving the core and quotient of the symbol. We list some applications, for example to the well-known hook lengths in integer partitions. This leads in particular to a generalization of a relative hook formula for the degree of characters of the symmetric group discovered by G.The celebrated hook formula for the degrees of the irreducible characters of the finite symmetric groups has been a source of inspiration for several other degree formulas. In his work [2] on unipotent degrees in reflection groups G. Malle used d-symbols as labels, defined hooks in d-symbols and associated a length to a hook. With these he was able to prove a "hook formula" for the degrees. He also proved formulas involving suitable cores and quotients of symbols.

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This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two p-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for p-blocks of alternating groups (where the blocks must also have the same sign when p is odd), of double covers of alternating and symmetric groups (for p odd, and where we obtain crossover isometries when the blocks have opposite signs), of complex reflection groups G(d, 1, n) (for d prime to p), of Weyl groups of type B and D (for p odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks, in a way which should be of independent interest.

In a paper of 2003, B. Külshammer, J. B. Olsson and G. R. Robinson defined ℓ-blocks for the symmetric groups, where ℓ > 1 is an arbitrary integer. In this paper, we give a definition for the defect group of the principal ℓ-block. We then check that, in the Abelian case, we have an analogue of one of M. Broué's conjectures.