Abstract
We give a simple explicit formula for an arbitrary \( 3j \)-symbol for the Lie algebra \( {\mathfrak{gl}}_{3} \). The symbol is expressed as the ratio of values of hypergeometric functions with \( \pm 1 \) substituted for all arguments. Finding a \( 3j \)-symbol is essentially equivalent to the determination of an arbitrary Clebsch–Gordan coefficient for \( {\mathfrak{gl}}_{3} \). The coefficients are important in the quark theory of quantum mechanics.
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Notes
The accurate definition of \( s \) is as follows: Write the decomposition as \( V\otimes W=\sum\nolimits_{U}M_{U}\otimes U \), where \( M_{U} \) is some vector space called the multiplicity space. If \( \{e_{i}\} \) is a basis for \( M_{U} \) then \( U^{s}:=e_{s}\otimes U \).
The support \( \operatorname{supp}F \) of a function expressed as the sum of a power series is the set of exponent vectors in the series of monomials.
Below we will consider only the representations with \( m_{3}=0 \) and \( m^{\prime}_{3}=0 \).
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 717–735. https://doi.org/10.33048/smzh.2022.63.401
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Artamonov, D.V. Formulas for Calculating the \( 3j \)-Symbols of the Representations of the Lie Algebra \( {\mathfrak{gl}}_{3} \) for the Gelfand–Tsetlin Bases. Sib Math J 63, 595–610 (2022). https://doi.org/10.1134/S0037446622040012
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DOI: https://doi.org/10.1134/S0037446622040012