Wreath products and projective system of non Schurian association schemes

TL;DR

This paper investigates wreath products of association schemes, characterizes their automorphism groups, and explores conditions under which they are non-Schurian, providing new examples of complex association schemes and their primitive idempotents.

Contribution

It determines the automorphism group of wreath products and characterizes when they are non-Schurian, including analysis of iterated wreath products and their primitive idempotents.

Findings

Wreath product is Schurian iff both components are Schurian.

Large families of non-Schurian association schemes are constructed.

Explicit description of primitive idempotents in iterated wreath products.

Abstract

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, with an explicit description of primitive idempotents.