## Saturday, April 16, 2011

### Theory Update 79

Welcome to all our readers from KITP (yes, bloggers can see who visits their site). In the talk yesterday, Arkani-Hamed mentions the work of the mathematician Goncharov (welcome also to viewers from Brown University). So we see in this recent paper the introduction of the associahedra polytopes, on page 7. Goncharov is also interested in tropical geometry, which is pretty cool. On page 13 we find a special $3 \times 3$ matrix. Devadoss is not mentioned, but moduli spaces are discussed. Happy reading!

1. I can't help but recall an evening in Oxford in 2009, where I sat near Nima at dinner. I was trying to show the twistor guys how interesting the associahedra were, but Nima said that category theory was yucky and he didn't want to know anything about it. Guess he's changed his mind, huh.

2. So I decided to finally get around to computing the Hopf algebra idempotents for an extension of the symmetric group. I got some code running last night and it spits out vectors that look right mathematically but with no physical interpretation.

The thing is that to do exact computer calculations (at least the easy way) I need the group to be finite. But given any finite group, it's fairly simple for me to modify the program to work with it. If you have any ideas on this they would be appreciated.

3. The extension of the symmetric group I was working on is the permutation group on three pairs. That is, the two elements of a pair stay together but can swap. This is a 6x8 = 48-element finite group and is non Abelian but otherwise is like $P_3\times P_2^3$ (which is the finite group I was using at the time of the above comment, due to a programming error).

The result was that the primitive idempotents had real parts (which I think of as weak hypercharge) with relative values of 1,2 and 3. For $P_3$ the values were 1 and 2 so this is an improvement. The actual primitive weak hypercharge relative values are 0,1,2,3,4,6, that is, 0,1/3, 2/3, 1, 4/3, and 2.