Separating Invariants of Quivers

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The ability to recognise when a pair of objects differ only up to symmetry is an important skill both for humans and artificial intelligence. For example, humans can easily recognise two different images of the same object taken from different angles, and in our increasingly automated world it is desirable for computers to be able to perform this task equally well.Humans distinguish objects up to symmetry by focussing on properties of the image which do not change, such as angles and distances between points of interest. These are called the invariants of the image. The mathematical object which encodes the idea of symmetry is a group action. Many important mathematical problems really boil down to this: given a pair of objects, is there a group element mapping one on to the other? Invariant theory seeks to solve problems like this in a natural way, by describing properties of the objects which remain fixed when the group action changes the object. Most of the historical work on invariant theory focussed on attempts to describe "all" the possible invariants in a given situation. In the last 20 years or so a new trend in invariant theory has emerged, in which we try to describe so-called "separating sets". These are sets of invariants which are able to determine, just as well the complete set of invariants, whether two objects are the same up to symmetry.A quiver is a network of nodes, with arrows pointing between them. One might imagine a diagram in which each node represents a city, and two nodes are connected if there is a direct flight running between the two cities. A representation of a quiver is a way of associating mathematical objects to quivers. These representations have been at the forefront of algebra research since the 1970's, thanks to a remarkable result which says essentially that almost all of representation theory can be reduced to representations of quivers. Classifying quivers up to symmetry is a problem which is amenable to an invariant-theoretic approach, and much progress has been made over the years by describing all invariants of quivers. This project seeks to bring together the two ideas described in the previous two paragraphs: describing separating sets for invariants of quivers. The chief benefits of this approach is that separating sets are often smaller and easier to use than complete sets of invariants. Thus, with knowledge of how to describe separating sets of quivers in hand we could increase our knowledge of how to classify quivers up to symmetry, and in doing so increase our understanding of representation theory in general.