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

Abstract

An explicit description of the multiplicity space that describes occurrences of irreducible representations in a splitting of a tensor product of two irreducible representations of ${\mathfrak{gl}}_n$ is given. Using this description an explicit formula for an arbitrary $6j$-symbol for the algebra ${\mathfrak{gl}}_n$ is derived. The $6j$-symbol is expressed through a value of a generalized hypergeometric function.

Several typos are corrected. Examples of semiinvariants are added. An example of caculation of a $6j$-symbol is added

References in corpus (5)

• $3j$-symbols for representation of the Lie algebra ${\mathfrak{gl}}_3$ in the Gelfand-Tselin base
• Clebsh-Gordan coefficients for the algebra ${\mathfrak{gl}}_3$ and hypergeometric functions
• Classical $6j$-symbols for finite dimensional representation of the algebra ${\mathfrak{gl}}_3$
• Models of representations for classical series of Lie algebras
• Integrals involving triplets of Jacobi and Gegenbauer polynomials and some 3j-symbols of SO(n), SU(n) and Sp(4)