0/1-Polytopes related to Latin squares autotopisms

Abstract

The set LS(n) of Latin squares of order n can be represented in Rn3 as a (n−1)3-dimensional 0/1-polytope. Given an autotopism Θ=(α,β,γ)∈An, we study in this paper the 0/1-polytope related to the subset of LS(n) having Θ in their autotopism group. Specifically, we prove that this polyhedral structure is generated by a polytope in R((nα−l1α)⋅n2+l1α⋅nβ⋅n)(l1α⋅l1β⋅(n−l1γ)+l1α⋅l1γ⋅(nβ−l1β)+l1β⋅l1γ⋅(nα−l1α)), where nα and nβ are the number of cycles of α and β, respectively, and l1δ is the number of fixed points of δ, for all δ∈{α,β,γ}. Moreover, we study the dimension of these two polytopes for Latin squares of order up to 9.