2003/07/19 by Henriette Elvang, Elvang, Henriette, Predrag Cvitanović +4
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics #hep-th #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.hep-th/0307186
14 pages, 225 figures
arxiv created 2003/07/19 · openalex publication_date 2003/07/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.