Determinants of latin squares of a given pattern

Abstract

Cycle structures of autotopisms of Latin squares determine all possible patterns of this kind of design. Moreover, given any isotopism, the number of Latin squares containing it in their autotopism group only depends on the cycle structure of this isotopism. This number has been studied in for Latin squares of order up to 7, by following the classification given in. Specifically, regarding each symbol of a Latin square as a variable, any Latin square can be seen as the vector space associated with the solution of an algebraic system of polynomial equations, which can be solved using Gröbner bases, by following the ideas implemented by Bayer to solve the problem of n-colouring a graph. However, computations for orders higher than 7 have been shown to be very difficult without using some other combinatorial tools. In this sense, we will see in this paper the possibility of studying the determinants of those Latin squares related to a given cycle structure. Specifically, since the determinant of a Latin square can be seen as a polynomial of degree n in n variables, it will determine a new polynomial equation that can be included into the previous system. Moreover, since determinants of Latin squares of order up to 7 determine their isotopic classes, we will study the set of isotopic classes of Latin squares of these orders related to each cycle structure.