$3j$-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.