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Two new gradings of the Lie algebra sl(3,C) are the refined sl(2,C) and o(3) gradings. The grading in each case utilizes the sl(2,C), or o(3), weight together with a new additive modular label. A complete set of sl(3,C) representation basis states labeled by each to the two sets of additive quantum numbers is found. The newlabeling operator in both cases fails to commute with the cubic sl(3,C) Casimir operator, and hence mixes states of the contragredient representations (p,q) and (q,p) [ABSTRACT FROM AUTHOR]

Generating functions are calculated for polynomial tensors in the components of tensors of the one- and two-dimensional space groups pm and p4m. For tensors whose k vectors lie at rational points (denominator q) of the Brillouin zone, the problem is imaged by the corresponding problem for an appropriate finite group, and known methods are used for its solution. For tensors with continuous k the solution is obtained by letting q become infinite. [ABSTRACT FROM AUTHOR]

The construction of generating functions for multiplicities of irreducible representations of Lie superalgebras from generating functions for characters is examined, and applied in order to obtain polynomial tensors and branching rules. Problems arising because of the existence of typical and atypical representations are discussed in detail. The techniques are applied to osp(1,2), spl(1,2), osp(3,2), and osp(4,2). [ABSTRACT FROM AUTHOR]

The Demazure-Tits subgroup of a simple Lie group G is the group of invariance of ClebschGordan coefficients tables (assuming an appropriate choice of basis). The structure of the Demazure-Tits subgroups of A[sub n], B[sub n], C[sub n], D[sub n], and G[sub 2] is described. Orbits of the permutation action of the DT group in any irreducible finite-dimensional representation space of A[sub 2], C[sub 2], and Ge are decomposed into the sum of irreducible representations of the DT group. [ABSTRACT FROM AUTHOR]

Indices and anomaly numbers for representations of basic classical Lie superalgebras are defined, and their explicit expressions are derived in terms of Kac–Dynkin labels. Useful properties of indices and anomalies are determined, and several examples are given. A similar analysis is made for superindices and superanomalies, and it is demonstrated how all these objects form a helpful tool in decomposing tensor products or in constructing branching rules for representations. [ABSTRACT FROM AUTHOR]

Using generating function methods, branching rules for Sp(6) ⊃ Sp(2) × O(3) are derived. The branching rules suggest an integrity basis, or set of elementary permissible diagrams, in terms of which the subgroup basis states are defined; they correspond to vibrational, or Bohr-Mottelson type, states in the nuclear symplectic model. [ABSTRACT FROM AUTHOR]