$3j$-symbols for representation of the Lie algebra ${\mathfrak{gl}}_3$ in the Gelfand-Tselin base

Abstract

In the paper a simple explicit formula for an arbitrary $3j$-symbol for the Lie algebra ${\mathfrak{gl}}_3$ is given. More precise necessary conditions for non-vanishing of a $3j$-symbol are given, in the case when these conditions hold we give an explicit expression for a $3j$-symbol. It is expressed through a fraction of values of $A$-hypergeometric function when one substitutes $\pm 1$ instead of all it's arguments. The problem of calculation of an arbitrary $3j$-symbol is equivalent to the problem of calculation of an arbitrary Clebsh-Gordan coefficient for the algebra ${\mathfrak{gl}}_3$. These coefficients play an important role in quantum mechanics in the theory of quarks.

arXiv commentA typo in the main Theorem is corrected

References in corpus (2)

• Semigroups of $s{l}_3(\mathbb{C})$ tensor product invariants
• Clebsh-Gordan coefficients for the algebra ${\mathfrak{gl}}_3$ and hypergeometric functions

Cited by in corpus (2)

• Classical $6j$-symbols for finite dimensional representation of the algebra ${\mathfrak{gl}}_3$
• Calculation of $6j$-symbols for the Lie algebra ${\mathfrak{gl}}_n$