On the Schaper Number of Partitions

TL;DR

This paper explores the Schaper Number of partitions, providing characterizations for certain cases and demonstrating how this knowledge can improve the calculation of decomposition numbers in symmetric group representations.

Contribution

It extends the understanding of Schaper numbers, especially for p=2, and offers new characterizations and conditions to facilitate decomposition number calculations.

Findings

Complete characterization of partitions with Schaper number at least three for p=2

Complete characterization of partitions with Schaper number at least four for p=2

Necessary conditions and conjecture for Schaper numbers at least three for odd primes

Abstract

One of the most useful tools for calculating the decomposition numbers of the symmetric group is Schaper's sum formula. The utility of this formula for a given Specht module can be improved by knowing the Schaper Number of the corresponding partition. Fayers gives a characterisation of those partitions whose Schaper number is at least two. In this paper we shall demonstrate how this knowledge can be used to calculate some decomposition numbers before extending this result with the hope of allowing more decomposition numbers to be calculated in the future. For $p=2$ we shall give a complete characterisation of partitions whose Schaper number is at least three, and those whose Schaper number at least four. We also present a list of necessary conditions for a partition to have Schaper number at least three for odd primes and a conjecture on the sufficiency of these conditions.