29 Feb 2008
From: Jutho.Haegeman [snail] ugent.be
A am I PhD student from the University of Ghent. For my research, I need
the irreducible representations of the orthogonal group. The final goal
is to construct the excited states of the Gross Neveu model on the
lattice, which behave as rank r anti-symmetrical tensor reps of O(2N).
To do these calculations I follow the approach of chapter 5, and thus I
need the 6j and 3j symbols for the tensorial irreps of O(2N). As I am
not aware of computational software that can calculate these quantities,
I tried to program it myself.
I implemented a way to work with O(N) birdtracks (undirected lines, with
possible contractions) into Matlab, and was able to construct the
projection operators for the tensorial irreps. For this, I use the
method described in chapter 10 of your webbook, starting from the Young
Projectors of the U(n) irreps, I first project out all possible
contractions (by sandwiching the contraction operator between the two
young projectors). I guess my implementation is correct, as I am
possible to obtain the exact irrep dimensions (for general O(n)) of all
the irreps which are present in the decomposition of V^4. I still have
two questions:
1) My current implementation is relatively slow, to construct the
projection operators for the irreps of the rank 4 tensors I already need
a few minutes, for the rank 5 tensors a few hours. This is due to the
fact that i try to project out all possible contractions (starting with
the highest number) combined with all possible lower dimensional irreps
for the not contracted lines. Is there an intelligent way to reduce the
number of possible contractions.
2) I use the general method to construct the Young Projectors described in
section 9.4. Yet in chapter 10 you use the symmetric projection operators
from figure 9.1. This has the advantage that the final projection operators
are symmetric (hermitian). Is there also a general algorithm to construct
these Young Projectors from figure 9.1?