Hook removal operators on the odd Young graph

Christine Bessenrodt,Jørn B Olsson,Eugenio Giannelli

doi:10.5802/alco.42

Abstract

In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.

Highlights

Let n be a natural number and let χ be an irreducible character of odd degree of the symmetric group Sn
: (i) We completely describe the partitions labelling those irreducible characters χ such that fkn−2k fkn(χ) = fkn+1(χ). (This description is obtained in Corollary 4.17 together with Lemma 4.11 and Proposition 4.13.)
The characters of odd degrees on which we focus in this article are labelled by partitions which we call odd partitions; we focus on a subgraph of the Young graph which we call the odd Young graph

Summary

Introduction

Let n be a natural number and let χ be an irreducible character of odd degree of the symmetric group Sn. In [8] it was shown that whenever k is a non-negative integer such that 2k n, there exists a unique irreducible constituent fkn(χ) of χSn−2k of odd degree appearing with odd multiplicity. Let Irr (Sn) be the set of irreducible characters of Sn of odd degree. In order to formulate the precise statements of our theorems we need to introduce some specific combinatorial concepts and notation. (ii) As a corollary of (i), we give an explicit formula for the number of such irreducible characters. The hook removal operators that we discuss here act on this odd Young graph, and the main theme of this article is the relationship of the behavior of these operators on this graph

Background

Preliminary results

Odd partitions

Commutativity