Calculation of $6j$-symbols for the Lie algebra $\mathfrak{gl}_n$

We explicitly construct linear generators for the multiplicity space that describes the occurrences of an irreducible representation in the decomposition of the tensor product of two irreducible finite-dimensional representations of the Lie algebra of all matrices of a given size. This result is applied to derive an explicit formula for an arbitrary $6j$-symbol associated with finite-dimensional representations of the Lie algebra. The value of such a symbol is expressed in terms of a generalized hypergeometric function.