Generalized hook lengths in symbols and partitions

TL;DR

This paper introduces a unified framework for generalized hook lengths in symbols and partitions, extending classical results and providing new formulas for character degrees in symmetric groups.

Contribution

It presents a comprehensive description of hooks in d-symbols, introduces generalized hook length functions, and generalizes a known hook formula for symmetric group characters.

Findings

Generalized hook length functions for d-symbols are established.

A new formula for character degrees of symmetric groups is derived.

Applications to integer partitions and representation theory are demonstrated.

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. Malle and G. Navarro in [3].