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Products and symmetrized powers of irreducible representations of Sp(2n, R) and their associates
No full textThe calculation of Kronecker products and plethysms of the infinite-dimensional harmonic series unitary irreducible representations of the non-compact group (Sp (2n, r) is considered. The complementarity of (Sp (2n, r) and O(k) is used to define associate irreducible representations of (Sp (2n, r). This leads to simple relationships between Kronecker products and plethysms of irreducible representations of (Sp (2n, r) and those of their corresponding associate irreducible representations. In the process of proving the validity of these previously conjectured relationships several new identities are found for plethysms involving infinite series of Schur functions. In addition, a general formula for plethysms of arbitrary irreducible representations of (Sp (2n, r) is derived and its mplementation is illustrated with a detailed example. A remarkable analogy is then observed between plethysms of the basic harmonic irreducible representations of (Sp (2n, r) and those of the basic spin irreducible representations of SO(2n)
Multiplicity free tensor products of irreducible representations of the exceptional Lie groups
No full textFor each of the exceptional Lie groups, a complete determination is given of those pairs of finite-dimensional irreducible representations whose tensor products (or squares) may be resolved into irreducible representations that are multiplicity free, i.e. such that no irreducible representation occurs in the decomposition of the tensor product more than once. Explicit formulae are presented for the decomposition of all those tensor products that are multiplicity free, many of which exhibit a stability property
The place of the adjoint representation in the Kronecker square of irreducible representations of simple Lie groups
No full textThe multiplicity of occurrence of the adjoint representation in the decomposition of the square of any finite-dimensional irreducible representation of any compact simple Lie group is shown to be equal to the number of non-vanishing components of the Dynkin label of . The resolution of this multiplicity into contributions to the symmetric and antisymmetric squares of is discussed, with complete results being found for all of the classical and some of the exceptional simple Lie groups, and partial results culminating in conjectures for the remaining exceptional groups
The square of the Vandermonde determinant and its q-generalisation
No full textThe Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlin's wavefunction ansatz. Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail. Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant. Algorithms for the efficient calculation of the q-dependent coefficients as finite polynomials in q are developed. The properties, such as the factorization of the q-dependent coefficients, are exposed. Further light is shed upon the vanishing of certain expansion coefficients at
q = 1. The q-generalization of the sum rule for the squares of the coefficients is derived. A number of compelling conjectures are stated
A finite subgroup of the exceptional Lie group G2
No full textWith a view to further refining the use of the exceptional group G2 in atomic and nuclear spectroscopy, it is confirmed that a simple finite subgroup L168~PSL2(7) of order 168 of the symmetric group S8 is also a subgroup of G2. It is established by character theoretic and other methods that there are two distinct embeddings of L168 in G2, analogous to the two distinct embeddings of SO(3) in G2. Relevant branching rules, tensor products and symmetrized tensor products are tabulated. As a stimulus to further applications the branching rules are given for the restriction from L168 to the octahedral crystallographic point group O
Products and symmetrized powers of irreducible representations of SO*(2n)
No full textThe calculation of branching rules, tensor products and plethysms of the infinite-dimensional harmonic series unitary irreducible representations of the non-compact group is considered and the duality between and Sp(2k) exploited. The branching rule for the restriction of an arbitrary harmonic series irreducible representation of to U(n) is derived, and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations, , of whose direct sum constitutes the metaplectic representation, H, of . A concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given. A general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of . Finally, relationships that arise from the embedding of the product groups and in the metaplectic group Mp(4nk) are discussed
Analogies between finite-dimensional irreducible representations of SO(2n) and infinite-dimensional irreducible representations of Sp(2n,R) Part I: Characters and products
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