Generalized hook lengths in symbols and partitions

Abstract

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.

Coefficients of Wronskian Hermite polynomials

“…Remark As mentioned at the end of Section 1, both corollaries also follow from theorem 4.4 in Ref. as mentioned in remark 4.5. In Ref.…”

“…Both corollaries also follow from a more general statement about hooks in partitions, see theorem 4.4 and remark 4.5 in Ref. 39, although our argument to arrive at these statements is completely different. Other related results about hook ratios can be found in Ref.…”

“…In Ref. 39, the inclusion of multisets of hook lengths is used, whereas we use the integrality of the coefficients of certain polynomials. Additionally, theorem 4.4 in Ref.…”

“…Then the transition from (̃, ) to ( , ) is represented by a dot moving from a position t to a position + 1 in the Maya diagram of . It is easy to see that both terms in (39) change parity in this transition if and only if ∉ + ′ . Hence, does not change in parity.…”

On bar lengths in partitions

“…In Section 3, we introduce a number of subsets of the set of hooks in the doubled partition, and derive from Macdonald's construction a number of properties of hook lengths and bar lengths. In Section 4, we then apply the results of [1] to deduce our main result, Theorem 4.1. We then finally apply the theorem to give a d-version of the bar formula (a relative bar formula) .…”