From Jutho.Haegeman Mar 3 2008
Subject: Re: so(n) 3-j and 6-j coefficients
I'm very interested in better (and faster) ways to calculate
the 3-j and 6-j coefficients. Does your implementation also
works for spinorial irreps? How slow/fast does it work? And
what language is it written in? My implementation is written in
Matlab. It does work for general O(n). For doing symbolical
calculations, Matlab communicates with Maple, which has another
negative influence on the overall speed. Fixing to a specific
n-value makes my code faster, yet the scaling to higher weight
irreps still remains and is of course the most terrible aspect
of the code.
I have two questions concerning the non-unitarity of the
projectors:
Does the remark of Georges Bergdolt about the non-orthogonality
(due to the non-unitarity) of the Young projectors pose a
problem? I start with a U(n)/GL(n) Young projector, and then
project out all possible traces, starting with the highest
number of contractions, and combining these with all possible
Young projectors for the free lines. (I reasoned that using the
Young projector or the real O(N) projector for the free lines
doesn't matter, as all higher number of contractions are
already projected out.)
I still have to write the code to calculate the 3-j and 6-j
coefficients. I first have to think trough which coefficients
I'd really need. I guess the major lines will be parallel with
the approach for U(n). Yet some possible difficulty already
came to mind.
In a U(n) Clebsch-Gordan series, you always end up with irrep
with a total number of boxes (in the Young diagram) equal to
the sum of the boxes of the two composing irreps. So the
Clebsch-Gordan birdtrack just consists of the three projectors
tied up in the correct order. In a O(N) Clebsch-Gordan series,
the number of boxes can decrease (due to contractions).
How then do you know which lines to contract? Probably this
information is in the Littlewood-rule. It uses S-functions,
which I do not know. And are such Clebsch-Gordan coefficients,
constructed from non-unitary projectors, unitary? Probably not.
How then to construct the complete projector, is it just
P1 P1
P
P2 P2
summed over all the irreps (with projector P) in the
decomposition, with added contractions for irreps with a lesser
number of boxes?
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