Classification of autoparatopisms of Latin cubes

Abstract

An action on a Latin hypercube of dimension d and order n is called a paratopism if the action is an element of the wreath product Sn≀Sd+1. A paratopism is said to be an autoparatopism if there exists a Latin hypercube, which is mapped to itself under the action of the paratopism. Given a Latin cube (when d = 3) and for each order n, (n ∈ Z+), up to the conjugacy in Sn≀S4 a classification is presented. It is proved that given an autoparatopism σ∈Sn≀Sd+1, every conjugate of σ is an autoparatpism. The most significant consequence for the process of classification is, if σ1 = (α1,α2,α3,α4;δ1) ∈ Sn≀S4 and σ2 = (β1,β2,β3,β4;δ2) ∈Sn≀S4 then, σ1 is conjugate to σ2 in Sn≀S4if and only if there is a length preserving bijection η from the cycles of δ1 to the cycles of δ2 such that if η maps a cycle (a1...ak) to (b1...bk) then αa1αa2...αak∼βb1βb2...βbk. As a consequence, it can be concluded that every autoparatopism σ1 is conjugate to another autoparatopism σ2 which is of one of the forms (α1,α2,α3,α4;ε), (ε,α2,α3,α4;(1 2)), (ε,ε,α3,α4;(1 2 3)), (ε,ε,ε,α4;(1 2 3 4)) or (ε,ε,α3,α4;(1 3)(2 4)).