Generalized Hook Lengths in Symbols and Partitions

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. Malle and G. Navarro in [3]. 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. Considering here for simplicity an h-hook as a pair (a, b) of integers satisfying 0 ≤ b < a and a − b = h, an h-hook may contribute a factor h (for characters of the symmetric groups), or qh − 1 (for unipotent characters in general linear groups over GF (q)), or more generally qh − for some complex root of unity to the degree of a character. Thus knowledge of the number of h-hooks in a partition or a symbol for all h gives information about character degrees. This paper is concerned with ways of organizing hooks which may lead to alternative versions of hook formulas. The hope is that these versions may be better suited to deal with explicit degree problems. We introduce generalized hook length functions for d-symbols and prove a general result about them. More specifically we consider certain functions h from the set H(S) of hooks of a d-symbol S to R and decompositions of Date: November 10, 2011. 2000 Mathematics Subject Classification 20C30, 20C33, 05A17