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when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions, vol. 3, page 382.

As far as I can see, $l_{s,n}-l_{k,n-1}$ can turn to zero, if $k=s$ and the elements of the Gelfand-Zeitlin pattern $m_{s,n}=m_{s,n-1}=0$, which gives (-1)! which is undefined. What am I overlooking here?

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Can you provide eq. (6)? I don't have the reference at hand. And what are $G_1$ factors? –  draks ... May 14 '12 at 17:56

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