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Category O and hexagons

Category O blackboard

These beautiful blackboard diagrams were drawn by Rollo Jenkins of Edinburgh University during the first year of his PhD, when he was trying to understand a mathematical object called Category O. This object contains a collection of nicely behaved modules related to 3×3 matrices with trace zero.

For each module in Category O there is an associated pair of numbers called a weight, and each weight can be thought of as a single point in a diagram like the one in the top-right of the photo. The geometry of the diagram completely determines the behaviour of the modules.

One thing you notice looking at the green/yellow picture in the top-right is the hexagonal pattern of vertices. It turns out that if you look at 6 of the green/yellow vertices which are in a hexagon, then these 6 modules alone tell you everything you need to understand about all of the modules. But why is it hexagons that are important and not some other shape? Rollo has noticed that if you look at the following matrix with zeros on the diagonal then the non-zero numbers appear to form a hexagon if you tilt your head the right way. Coincidence?

Trace zero 3x3 matrix
The 6 stars (non-zero entries) form a hexagon. Coincidence?

(Rollo did all this work in the computer room instead of working in his own office. Two years later nobody has had the heart to remove the diagrams!)

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