Published on September 10th, 2013 | by Daphane Ng0
Self-similar actions of groups on graphs
By Lachlan Macdonald, University of Wollongong
This student took part in the 2012/13 AMSI Vacation Research Scholarship program. For more information on this year’s program please click here.
The symmetries of an object are the ways in which we can shift an object around so that at the end, it looks the same as how it started. In mathematics, we use “groups” to model the symmetries of objects.
As an example, the symmetries of an equilateral triangle are rotation of the triangle by 120 degrees (so the corners match up), and flipping the triangle about any one of its three axes of symmetry. Any combination of these flips and rotations is also a symmetry; for example, flipping the triangle once about its vertical axis of symmetry, and rotating it 120 degrees gives us back the triangle looking identical to how it did before we flipped and rotated it. Even simply leaving the triangle alone is a symmetry, and it’s a symmetry that can also be achieved by flipping about an axis of symmetry twice, or rotating the triangle three times in the same direction. “Groups” in mathematics give us a way of encoding these properties, and allow us to extend the notion of “symmetry” to more abstract objects and work in a more general framework.
One way groups are used is to model the symmetries of an abstract mathematical object called a “graph”. Some familiar examples of graphs include the trees we use to visualise outcomes in probability theory. For example, in visualising the outcomes of throwing a coin, we have two arrows, one for heads and one for tails, branching out from a single starting point. Each arrow then hits another point, where it branches off into another heads arrow and tails arrow, and so on. In graph theory, we refer to each individual “arrow” as an “edge”, and we refer to the nodes that an edge starts or ends at as “vertices”. Graphs like these and others also have symmetries – and lots of them, and while some of these symmetries can be tricky to visualise, we can still model them all and explore their properties using groups. A “self-similar action” of a group on a graph is a way of modelling a special type of symmetry of a graph.
We can make use of the information we have about a symmetry by what’s known as “representation theory”. Representation theory allows us to work with symmetries in infinite dimensional space, giving us notions of addition and multiplication. Self-similar actions in particular give us some very useful structure in this framework, with applications to quantum physics in modelling the quantum states of physical systems.