Basis states and generator matrix elements are given for the generic representation (a, b) of G 2 in an SU (3) basis.

We construct SL(3,C) basis states reduced according to its finite subgroup ℘3. Matrix elements of sl(3,C) generators are calculated between ℘3 basis states.

Complete branching rules for B-n superset of Bn-1 x U(1) and D-n superset of Dn-1 x U(1) are given in generating function form.

We present an example of calculation of affine Weyl orbit to orbit branching rules. The results are presented in terms of an integrity basis and compatibility rules.

Generating functions are found for decomposition of the space of homogeneous polynomials of any degree in four variables into the direct sum of subspaces irreducible under the group H-4, the non-crystallographic Coxeter group of order 14400. The four variables are coordinates of a vector from the defining four-dimensional representation space of the group. As the defining representation we consider any of the four non-equivalent irreducible representations of H-4 of dimension four.Analogous generating functions for the binary icosahedral group H-3 of order 120 (generated by reflections, i.e. a subgroup of O(3) but not of SO(3)) and for the dihedral group H-2 of order 10 are also rederived and shown. The groups are naturally related by H-4 superset of H-3 superset of H-2.

Complete orbit–orbit branching rules are derived for each classical algebra–maximal-subalgebra pair Cm+nCm⊕Cn, Bm+nDm⊕Bn, Dm+nDm⊕Dn. Since each pair is equal rank, and algebra and subalgebra Weyl sectors line up, the integrity basis in each case consists of the subalgebra orbits contained in the fundamental orbits of the algebra.

A new set of polynomial states (to be called character states) are derived for Sp(4) reduced to its SU(2) × U(1) subgroup, and the relevant generator matrix elements are evaluated for generic representations (a,b) of Sp(4). (The degenerate representations (a,0) and (0,b) were treated in our previous work and are also given in this paper). The group–subgroup in question is that of the seniority model of nuclear physics.

Complete orbit–orbit branching rules are found for the classical algebra–maximal reductive subalgebra pairs Am+n+1⊃Am⊕An⊕u(1), Bm+1⊃Bm⊕u(1), Cm+1⊃Am⊕u(1), Dm+1⊃Dm⊕u(1), and Dm+1⊃Am⊕u(1).

A new method of explicit description of n-dimensional polytopes generated by reflections of a single point (D-polytopes) or a face of maximal dimension (V-polytopes) is used to provide a comprehensive uniform description of all such polytopes in dimensions three and four. In addition a class of six-dimensional polytopes is also described; its symmetry group is the Weyl group of the simple Lie group D-6.

Basis states and generator matrix elements are given for the generic representation $(a,b)$ of $G_2$ in an $SU(3)$ basis.

Basis states and generator matrix elements are given for the generic representation $(a,b)$ of $G_2$ in an $SU(3)$ basis.

Basis states and generator matrix elements are given for the degenerate representations (a, 0) and (0, b) of G2 in an SU(3) basis. We show that for any compact Lie group the elementary unwanted states, and hence incompatibilities between fundamental basis states in the character generator, are all of degree 2.

It is shown how the derivation of group-subgroup (IR to IR) branching rules is facilitated by the use of the much simpler (Weyl) orbit-orbit branching rules.

Complete orbit–orbit branching rules, or equivalently, reduction of Weyl group orbits, between each simple algebra of rank 3 or less, and its equal-rank subalgebras and between F4 and each of its equal-rank subalgebras are given. The generic case An⊇An−1×U(1), the subjoining F4≳B3×U(1), and E8⊇E6×A2 (first and seventh highest weight labels nonzero) are also treated. Branching rules between rank 4 and 5 simple algebras and E6 and their equal-rank subalgebras are available in a depository.

Weyl orbits of affine rank 2 and 3 algebras are obtained analytically, along with the depths of their weights, in terms of an integrity basis consisting of sets of compatible (or adjacent) weights, one from each fundamental orbit. The fundamental orbits and others of the same level and congruence class are then decomposed into irreducible representations (IR). Since the orbit →IR matrix is triangular, it will be easy to invert it to get orbit multiplicities and subalgebra branching rules.

The Wigner–Inönü contraction is generalized to include gradings other than Z2 and the formalism is applied to obtain all toroidal contractions of sl(3,C). Examples of contractions based on nontoroidal gradings are also given.

We construct basis states for SU(3) and for SO(5) that are polynomials in the states of the fundamental representations; they are reduced according to the finite Demazure-Tits subgroup, which acts on basis states in the same manner that the Weyl group acts on weights.

See ibid., vol.22, p.L191 (1989). In a recent letter on the generalised exponents of the Lie algebra sl(3,C), Scutaru refers to an integrity basis of the enveloping algebra proposed by the authors as a hypothesis and then proceeds to establish its validity. The purpose of this comment is to stress the fact that this integrity basis was not a hypothesis but a proven result.

When constructing polynomial bases for group representations, one is led naturally to consider a 'stretched product' of states. Elementary multiplets are states that cannot be expressed as the stretched product of any other two states, and they can be identified with terms occurring in certain generating functions. In this paper the authors define an analogous stretched product of Young tableaux together with elementary tableaux. These may be used in a manner similar to elementary multiplets to construct generating functions. They consider several examples where this method can be applied and in particular give new large-N (rank-free) branching rule generating functions for the group-subgroup pairs SU(n)*SU(m) contained in/implied by SU(n+m) (first four labels non-zero), SO(n) contained in/implied by SU(n) (first three labels non-zero) and Sp(2n) contained in/implied by SU(2n) (first five labels non-zero). This method of finding these generating functions uses the theory of Grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux.