Paul Martin's Algebra Projects Links Pages

Diagram algebra/category project (Summer)

A diagram category is a rather beautiful kind of generalisation of the idea of a finite group, such as the symmetric group. Like groups, these algebras arise in Physics. But while groups describe the symmetries in Physical systems, diagram categories describe the (mathematical) model of the systems themselves.
An overarching example of a diagram category is the partition category described here . Just like the symmetric groups have many subgroups, so the partition category has many subcategories. If one puts aside the Physical applications for a moment, then it becomes a challenging exercise to try to find new subcategories with interesting structures. Many such substructures have been found. But there are certainly many more still to be discovered!
Project themes:

What are they?
How could they be explained to a wide audience of clever non-algebraists?
How can their "pictorial" nature be used in rigorous calculations?
How do they connect with their applications?
... Can you find a new one!?

Keywords: Doing algebra with pictures
Links:

On Diagram Categories ... (preprint version of article in AMS Contemporary Mathematics)
for Simon.
for James (old).
to follow

(meanwhile, if you are interested, just ask me!).
For some clues about the connections to Physics see Chapter 11 here Potts models and related problems in statistical mechanics.