Ronald C. King

Primitive vectors of Kac-modules of the Lie superalgebras sl(m/n)

It was conjectured by Hughes et al. [J. Math. Phys. 33, 470–491 (1992)] that there exists a bijection between the composition factors of a Kac-module and the so-called permissible codes. In a previous paper it was proved that to any unlinked code, there corresponds a composition factor of the Kac-module. Here it is proved that to any linked code, there corresponds a composition factor of the Kac-module.

The square of the Vandermonde determinant and its q-generalization

The Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlins 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.

Factorisation of Littlewood--Richardson coefficients

The hive model is used to show that the saturation of any essential Horn inequality leads to the factorisation of Littlewood-Richardson coefficients. The proof is based on the use of combinatorial objects known as puzzles. These are shown not only to account for the origin of Horn inequalities, but also to determine the constraints on hives that lead to factorisation. Defining a primitive Littlewood-Richardson coefficient to be one for which all essential Horn inequalities are strict, it is shown that every Littlewood-Richardson coefficient can be expressed as a product of primitive coefficients. Precisely the same result is shown to apply to the polynomials defined by stretched Littlewood-Richardson coefficients.

A finite subgroup of the exceptional Lie group G2

With 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.

Symmetrised powers of rotation group representations

The results given permit the unambiguous evaluation of all possible Kronecker products of the irreducible representations (tensor and spinor) of On and SOn for n=2 nu and n=2 nu +1. A complete resolution of the second and third powers of the basic spinor representations of SO2 nu and SO2 nu +1 is given, together with a prescription for analysing the fourth power of these representations. Detailed application is made to the enumeration of properties of SO10 relevant to grand unified theories, and sufficient information given to resolve the fourth power of any representation of SO10.

The Hopf algebra structure of the character rings of classical groups

The character ring \CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \Sym

A non-commutative n-particle 3D Wigner quantum oscillator

of symmetric functions. Here we study the character rings \CO and \CSp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that \CO and \CSp also admit natural Hopf algebra structures that are isomorphic to that of \CGL, and hence to \Sym. The isomorphisms are determined explicitly, along with the specification of standard bases for \CO and \CSp analogous to those used for \Sym. A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the \CGL case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras \CO and \CSp are not self-dual. The dual Hopf algebras \CO^* and \CSp^* are identified. Finally, the Hopf algebra of the universal rational character ring \CGLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

Analogies between finite-dimensional irreducible representations of SO(2n) and infinite-dimensional irreducible representations of Sp(2n,R). II. Plethysms

An n-particle three-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigners suggestion that Hamiltons equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p ≥ 3n but atypical for p < 3n. The branching rules for the restriction from sl(1|3n) to gl(1) ⊕ so(3) ⊕ sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n ≤ 2. It is shown that measurements of the coordinates of the individual particles give rise to a set of discrete values defining nests in the three-dimensional configuration space. The fact that the underlying geometry is non-commutative is shown to have a significant impact on measurements of particle separation. In the atypical case, exclusion phenomena are identified that are entirely due to the effect of A-superstatistics. The energy spectrum and associated degeneracies are calculated for an infinite-dimensional realization of the Wigner quantum oscillator model obtained by summing over all p. The results are compared with those applying to the analogous canonical quantum oscillator.

Young diagrams, supercharacters of OSp(M/N) and modification rules

The basic spin difference character Δ″ of SO(2n) is a useful device in dealing with characters of irreducible spinor representations of SO(2n). It is shown here that its kth-fold symmetrized powers, or plethysms, associated with partitions κ of k factorize in such a way that Δ″⊗{κ}=(Δ″)r(κ)Πκ, where r(κ) is the Frobenius rank of κ. The analogy between SO(2n) and Sp(2n,R) is shown to be such that the plethysms of the basic harmonic or metaplectic character Δ of Sp(2n,R) factorize in the same way to give Δ⊗{κ}=(Δ)r(κ)Πκ. Moreover, the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of SO(2n) and Sp(2n,R) not only for the plethysms themselves, but also for their factors Πκ and Πκ. Explicit formulas are derived for each of these decompositions, expressed in terms of various group–subgroup branching rule multiplicities, particularly those defined by the restriction from O(k) to the symmetric group Sk. Illustrative examples are included, as well as a...

The evaluation of weight multiplicities using characters and S-function

The supercharacter of OSp(M/N) associated with an arbitrary Young diagram is defined. The distinction is made between OSp(M/N) standard and non-standard supercharacters. The corresponding modification rule which may be used to express a non-standard supercharacter in terms of standard supercharacters is presented, exemplified and proved. This rule involves the removal of N/2+1 boundary strips from the Young diagram. In the case N=0 the rule reduces to the well known rule appropriate to O(M). For a non-standard supercharacter corresponding to a typical irreducible representation of OSp(M/N) the modification yields a single typical standard supercharacter. On the other hand, for a non-standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters.