On principal hook length partitions and durfee sizes in skew characters

TL;DR

This paper explores the relationships between principal hook lengths and Durfee sizes in skew characters, providing methods to identify partitions with extremal properties and conditions for skew diagram equivalence.

Contribution

It introduces a construction for partitions with maximal principal hook lengths and minimal Durfee sizes in skew characters, and characterizes when two skew diagrams represent the same character.

Findings

Identifies partitions with maximal principal hook lengths in skew characters.

Establishes minimal Durfee size for these partitions.

Provides maximal Durfee size for specific skew diagram cases.

Abstract

In this paper we construct for a given arbitrary skew diagram A all partitions nu with maximal principal hook lengths among all partitions with the character [nu] appearing in the skew character [A]. Furthermore we show that these are also partitions with minimal Durfee size. This we use to give the maximal Durfee size for [nu] appearing in [A] for the cases when A decays into two partitions and for some special cases of A. Also this gives conditions for two skew diagrams to represent the same skew character.