Speaker: Nick Cavenagh 
Affiliation: University of Waikato, NZ


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.

About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 2 to 3pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Ramiro Lafuente.


Priestley Building (67)
Room: 442