21-25 August 2012

Portorož, Slovenia

UTC timezone

# 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