A general framework for non-abelian symmetries is presented formatrix-product and tensor-network states in the presence of orthonormal localas well as effective basis sets. The two crucial ingredients, theClebsch-Gordan algebra for multiplet spaces as well as the Wigner-Eckarttheorem for operators, are accounted for in a natural, well-organized, andcomputationally straightforward way. The unifying tensor-representation forquantum symmetry spaces, dubbed QSpace, is particularly suitable to deal withstandard renormalization group algorithms such as the numerical renormalizationgroup (NRG), the density matrix renormalization group (DMRG), or also moregeneral tensor networks such as the multi-scale entanglement renormalizationansatz (MERA). In this paper, the focus is on the application of thenon-abelian framework within the NRG. A detailed analysis is given for a fullyscreened spin-3/2 three-channel Anderson impurity model in the presence ofconservation of total spin, particle-hole symmetry, and SU(3) channel symmetry.The same system is analyzed using several alternative symmetry scenarios. Thisincludes the more traditional symmetry setting SU(2)^4, the larger symmetrySU(2)*U(1)*SU(3), and their much larger enveloping symplectic symmetrySU(2)*Sp(6). These are compared in detail, including their respective dramaticgain in numerical efficiency. In the appendix, finally, an extensiveintroduction to non-abelian symmetries is given for practical applications,together with simple self-contained numerical procedures to obtainClebsch-Gordan coefficients and irreducible operators sets. The resultingQSpace tensors can deal with any set of abelian symmetries together witharbitrary non-abelian symmetries with compact, i.e. finite-dimensional,semi-simple Lie algebras.
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