A Classification of Autoparatopisms of Latin Cubes

TL;DR

This paper classifies autoparatopisms of Latin cubes of dimension 3 and order n, analyzing their structure within the wreath product group and establishing conjugacy conditions based on cycle structure.

Contribution

It provides a classification of autoparatopisms of Latin cubes up to conjugacy, including proofs that conjugates of autoparatopisms are also autoparatopisms and conditions based on cycle structure.

Findings

All conjugates of an autoparatopism are autoparatopisms.

Cycle structure of the permutation determines conjugacy classes.

Classification up to conjugacy in the wreath product group.

Abstract

A paratopism is an action on a Latin hypercube of dimension d and order n which is an element of the wreath product $S_n \wr S_{d+1}$. A paratopism is said to be an autoparatopism if there is at least one Latin hypercube which is mapped to itself under the action of the paratopism. In this paper we classify autoparatopisms of Latin cubes given d = 3 and $n \in \mathbb{Z^+}$, upto the conjugacy in $S_n \wr S_4$. In order to achieve this objective, we prove that given an autoparatopism ${\sigma} \in S_n \wr S_{d+1}$, all the conjugates of ${\sigma}$ are autoparatpisms. Also, an important condition is presented, which states that the cycle structure of the ${\delta}$ in the element ${\sigma}= ({\alpha}_1,{\alpha}_2,{\alpha}_3,{\alpha}_4;{\delta})$ of $S_n \wr S_4$ determines the conjugacy.