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The exceptional symmetry
How many ways are there to choose two objects from a collection of six? 15. How many ways are there to sort six objects into three pairs of two? 15. Is there some deep significance to the fact that both these questions have the same answer? Yes!
This picture, which appears in a recent paper by Jon McCammond, shows the combinatorial object known informally as the doily. It is closely related to the two sets mentioned above. Ignoring the diagrams within the small circles for now, we see that the doily consists of fifteen points (the small circles) and fifteen lines. Some of the “lines” are drawn as curves in the picture, but each line links exactly three of the points together. The data specifying which points lie on which lines determines a point-line incidence structure, and this defines the structure of the doily.
The pictures inside each small circle lead to some further insights. Looking at the doily on the left, we see that the labels on the circles correspond to the 15 ways of choosing a pair of objects from a collection of six; the pair is indicated by the red line. There will be a line joining two of the points if and only if the points correspond to non-overlapping pairs. This means that the three points lying on a given line correspond to one of the 15 ways to sort six elements into three non-overlapping pairs, and it follows that there will be 15 such lines.
The doily can be constructed from the observation that a given point lies on a given line if and only if the pair that labels the point is one of the triple of pairs that labels the line. However, drawing the doily in an aesthetically pleasing way takes some skill.
In the version of the doily on the right, the roles of points and lines have been reversed, although the doily itself looks exactly the same. (Mathematicians express this by saying that the doily is self-dual.) In the right-hand version of the doily, the points are labelled by the 15 ways to decompose six objects into three pairs. Three of these decompositions will be joined by a line if and only if there is one pair common to all three of them. It follows that the lines are in correspondence with the 15 ways of choosing a pair from a collection of six objects.
The reason this is significant is that it illustrates a remarkable property about symmetry groups in abstract algebra. Given n objects, we can form the symmetric group Sn, which consists of all the n! (n factorial) ways of permuting the n objects, under the operation of composition. For example, if n=6, then the symmetric group S6 contains 6!=6x5x4x3x2x1=720 permutations. The group of symmetries Sn itself has symmetries, called automorphisms; these are the ways of permuting the symmetries in a way that respects the way they compose. (The automorphisms of a group are the isomorphisms from the group to itself.) Usually what happens is that the symmetries of Sn arise from a relabelling of the n objects being permuted; mathematicians call these inner automorphisms. But if n=6, Sn has additional automorphisms, called outer automorphisms, and this never happens for any other value of n.
The picture shows how to generate outer automorphisms. On the left, the points of the doily correspond to transpositions; that is, permutations of six objects that fix four of them and exchange the other two. On the right, the points of the doily correspond to triple transpositions; an example of one of these might simultaneously exchange point 1 with point 4, point 2 with point 6, and point 3 with point 5. The two versions of the picture can be used to define an automorphism of S6 that sends transpositions to triple transpositions.
Jon McCammond’s recent paper The Exceptional Symmetry (http://arxiv.org/abs/1412.1855) gives an elementary proof that this miracle involving the automorphisms of the symmetric group happens for n=6, but for no other value of n.
I’ve posted about this before, but it was two years ago, before many people read my posts. My other post (https://plus.google.com/101584889282878921052/posts/PJG9awS4MD5) goes into more detail about generalized quadrangles (of which the doily is the simplest nontrivial example) and J.J. Sylvester, who studied the combinatorics of the doily in 1844.
The nickname “doily” was invented by S.E. Payne in 1973. You can find out more about generalized quadrangles here: http://en.wikipedia.org/wiki/Generalized_quadrangle
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