Defect of characters of the symmetric group

TL;DR

This paper introduces a new concept of efect for characters of symmetric groups, providing a hook-length formula analogue and proving an onjecture for cases where n < efect^2.

Contribution

It defines the efect for symmetric group characters and establishes an nalogue of the hook-length formula, proving a version of the McKay Conjecture for certain cases.

Findings

efect is given by a hook-length formula analogue

Proves an onjecture when n < efect^2

Extends the theory of blocks in symmetric groups

Abstract

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 .