Association schemes with given stratum dimensions: on a paper of Peter M. Neumann

TL;DR

This paper revisits Neumann's unpublished 1969 work on primitive permutation groups of degree 3p, reinterpreting it as a combinatorial result on association schemes with specific eigenspace dimensions, independent of primality or permutation groups.

Contribution

It extracts and clarifies combinatorial results from Neumann's work, focusing on association schemes with prescribed eigenspace dimensions, without relying on primality or permutation group structures.

Findings

Association schemes with limited eigenspace dimensions are characterized.

The results are independent of the primality of p and permutation group existence.

The paper provides detailed combinatorial analysis of these schemes.

Abstract

In January 1969, Peter M. Neumann wrote a paper entitled "Primitive permutation groups of degree 3p". The main theorem placed restrictions on the parameters of a primitive but not 2-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree. However, there are further reasons for being interested in this paper. First, it was written at a time when combinatorial techniques were being introduced into the theory of finite permutation groups, and the paper gives a very good summary and application of these techniques. Second, like its predecessor by Helmut Wielandt on primitive groups of degree 2p, it can be re-interpreted as a combinatorial result concerning association schemes whose common eigenspaces have dimensions of a rather limited form. This result uses neither the primality of p nor the existence of a permutation group related to the combinatorial structure. We extract these results and give details of the related combinatorics.