Latin trades derived from Latin square automorphisms, autotpisms and autoparatopisms

Speaker: Nick Cavenagh 
Affiliation: University of Waikato, NZ

Abstract

Latin trades describe differences between Latin squares of the same order. In 2008, Cavenagh, Dr\'{a}pal et al showed that Latin trades can be constructed from groups with certain properties. This construction is ``nice'' in the sense that (a) the Latin trades are entry-transitive (a symmetry maps any ordered triple to any other ordered triple); (b) useful properties of the Latin trade such as minimality can be established using properties of the group; and (c) the construction yields examples of large and minimal trades that embed in group operation tables. However, the construction does not give a direct embedding of the Latin trade into any particular Latin square.

In this talk we generalize the above result to show that an action of the ``auto-equivalence'' group of a Latin square, can, with certain properties, yield Latin trades within that Latin square that again have ``nice'' properties. Auto-quivalences of Latin squares include, in an order increasingly general, {\em automorphisms}, where the same permutation is applied to rows, columns and symbols; {\em autotopisms}, where distinct permutations are applied to rows, columns and symbols; and {\em autoparatopisms} where the row, column and symbol sets are permuted set-wise, for example, the transpose.

 Applying this theory, we use Mersenne primes $2^q-1$ to locate trades in the group table for $({\mathbb Z}_2)^q$ and we identify trades in Latin squares that arise from quadratic orthomorphisms.