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Re: Quantum Gravity Seminar: Week 7: Track 2

Paul Arendt wrote, concerning
the diagrammatic approach to Clebsch-Gordan coefficients:

>This is absolutely mega-mega cool!  

>Somebody asked a short while ago
>on s.p.r. about how to derive this formula, using polynomials instead
>of the diagrammatic notation.  I was going to rattle off (what would
>probably have been) some barely intelligible garbage about
>highest-weight decomposition and conventions for Clebsch-Gordan
>coefficients, while not knowing how to do it the polynomial way.
>The diagrams make it seem much, much, much more sensible!

Yes, diagrams are often the way to go for this stuff.

>But I must ask: does the diagrammatic technique generalize to
>other (semisimple?) Lie groups?  A quick guess: only if there's
>a unique fundamental representation?

Good guess!

The method of diagrams works for studying representations of any
group, but the more you know about the group and its representations,
the more you can do.   Apparently the Wizard will be talking more 
about this as the course proceeds.  He says he started with SU(2) for 
some very good reasons: to keep things simple, to connect the subject 
to what folks already know about the quantum mechanics of angular 
momentum, and because SU(2) is important for quantum gravity.

As you guessed, the method of diagrams works especially nicely when your
group has a representation V such that every finite-dimensional 
irreducible representation can be produced by taking tensor powers 
of V and then applying various "mixtures" of symmetrization and 
antisymmetrization.  To describe these "mixtures" we need to use 
"Young diagrams", as sketched here:


All the "classical" Lie groups work this way, e.g. SL(n,R), SL(n,C),
SU(n), SO(n), and Sp(n).   In every case we can use the obvious 
n-dimensional representation as our representation V.  Folks call
this the "fundamental" representation.  It's usually not the *unique*
representation with the above property, but that's okay. 

The case of SU(2) is even nicer, because in this case, to get all 
the irreducible representations, all we need is *symmetrized* tensor 
powers of the fundamental rep V = C^2.  So we don't even need Young 

The same is true for SO(3), of course.

You can learn more about this stuff from:

Geoffrey E. Stedman, Diagram techniques in group theory, Cambridge,
Cambridge University Press, 1990.

and also a really cool book by a guy named something like Ctanovic, 
which is presently almost impossible to find unless you're a wizard.  
(This book will eventually be published more widely, after what's-his-name
inserts a section on diagram techniques for exceptional Lie groups!)