# Computers in Scientific Discovery 6

21-25 August 2012
Portorož, Slovenia
UTC timezone
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# SYMMETRIES OF LATIN SQUARES

Presented by Ian WANLESS
Type: Keynote Lecture

## Content

A Latin square of order $n$ is an $n\times n$ matrix containing $n$ different symbols (often the numbers $1,\dots,n$), positioned in such a way that each symbol occurs exactly once in each row and exactly once in each column. Latin squares have numerous applications in design of experiments, schedules for sporting tournaments, codes for communication and so on. In this talk I will consider, from a pure mathematical perspective, the possible symmetries of Latin squares. Consider the following examples: $A=\begin{pmatrix} 3&6&1&2&5&4\\ 4&1&2&3&6&5\\ 5&2&3&4&1&6\\ 6&3&4&5&2&1\\ 1&4&5&6&3&2\\ 2&5&6&1&4&3\\ \end{pmatrix} \qquad B=\begin{pmatrix} 1&6&5&2&3&4&7\\ 6&2&7&3&5&1&4\\ 5&7&3&1&4&6&2\\ 2&3&1&4&7&5&6\\ 7&4&2&6&1&3&5\\ 3&5&4&7&6&2&1\\ 4&1&6&5&2&7&3\\ \end{pmatrix}$ The square $A$ has an obvious pattern whereby the numbers occur in the same cyclic order in each column. Mathematically, we capture this idea by saying it has a symmetry that applies the cyclic permutation $(123456)$ to the rows, and the same permutation to the symbols. The square $B$ has a less obvious symmetry that applies the permutation $(123)(4)(567)$ to the rows, columns and symbols at the same time. Given $3$ permutations $\alpha,\beta,\gamma$, does there exist a Latin square which has a symmetry whereby $\alpha$ is applied to the rows, $\beta$ is applied to the columns and $\gamma$ is applied to the symbols, but overall the square is unchanged? If so then $(\alpha,\beta,\gamma)$ is called an {\em autotopism}. The special case when $\alpha=\beta=\gamma$ (as it did in example $B$ above), is known as {\em automorphism}. I will consider the question of which permutations can be used to build autotopisms and automorphisms. The (partial) answers that I provide will be a nice blend of theory and computation, with both aspects facilitating each other.

## Place

Location: Portorož, Slovenia
Address: University of Primorska, Faculty of Tourism Studies, Obala 11a, SI-6320 Portorož - Portorose, Slovenia
Room: VP1

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