Diagrammatic Young Projection Operators for U(n)

TL;DR

This paper introduces a diagrammatic method for constructing and analyzing Young projection operators for U(n), enabling the evaluation of invariant tensors and 3n-j coefficients with new sum rules and relations.

Contribution

It develops a diagrammatic approach to Young projection operators for U(n), proving their properties and deriving new formulas for invariant scalars and 3n-j coefficients.

Findings

Constructed diagrammatic Young projection operators for U(n).

Derived formulas for 3n-j coefficients and their proportionality to representation dimensions.

Established new sum rules and relations for 3-j and 6-j coefficients.

Abstract

We utilize a diagrammatic notation for invariant tensors to construct the Young projection operators for the irreducible representations of the unitary group U(n), prove their uniqueness, idempotency, and orthogonality, and rederive the formula for their dimensions. We show that all U(n) invariant scalars (3n-j coefficients) can be constructed and evaluated diagrammatically from these U(n) Young projection operators. We prove that the values of all U(n) 3n-j coefficients are proportional to the dimension of the maximal representation in the coefficient, with the proportionality factor fully determined by its S[k] symmetric group value. We also derive a family of new sum rules for the 3-j and 6-j coefficients, and discuss relations that follow from the negative dimensionality theorem.