Trevor A. Welsh

The mixed two-qubit system and the structure of its ring of local invariants

The local invariants of a mixed two-qubit system are discussed. These invariants are polynomials in the elements of the corresponding density matrix. They are counted by means of group-theoretic branching rules which relate this problem to one arising in spin–isospin nuclear shell models. The corresponding Molien series and a refinement in the form of a four-parameter generating function are determined. A graphical approach is then used to construct explicitly a fundamental set of 21 invariants. Relations between them are found in the form of syzygies. By using these, the structure of the ring of local invariants is determined, and complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.

A computer code for calculations in the algebraic collective model of the atomic nucleus

A Maple code is presented for algebraic collective model (ACM) calculations. The ACM is an algebraic version of the Bohr model of the atomic nucleus, in which all required matrix elements are derived by exploiting the model’s SU(1,1)×SO(5) dynamical group. This paper reviews the mathematical formulation of the ACM, and serves as a manual for the code. The code enables a wide range of model Hamiltonians to be analysed. This range includes essentially all Hamiltonians that are rational functions of the model’s quadrupole moments qˆM and are at most quadratic in the corresponding conjugate momenta πˆN (−2≤M,N≤2). The code makes use of expressions for matrix elements derived elsewhere and newly derived matrix elements of the operators [πˆ⊗qˆ⊗πˆ]0 and [πˆ⊗πˆ]LM. The code is made efficient by use of an analytical expression for the needed SO(5)-reduced matrix elements, and use of SO(5)⊃SO(3) Clebsch–Gordan coefficients obtained from precomputed data files provided with the code.

Construction of SO(5)⊃SO(3) spherical harmonics and Clebsch–Gordan coefficients

The SO(5) ⊃SO(3) spherical harmonics form a natural basis for expansion of nuclear collective model angular wave functions. They underlie the recently-proposed algebraic method for diagonalization of the nuclear collective model Hamiltonian in an SU(1,1) ×SO(5) basis. We present a computer code for explicit construction of the SO(5) ⊃SO(3) spherical harmonics and use them to compute the Clebsch-Gordan coefficients needed for collective model calculations in an SO(3)-coupled basis. With these Clebsch-Gordan coefficients it becomes possible to compute the matrix elements of collective model observables by purely algebraic methods.

Qubits and invariant theory

The invariants of a mixed two-qubit system are discussed. These are polynomials in the elements of the corresponding density matrix. They are counted by means of grouptheoretic branching rules and the Molien function is determined. The fundamental invariants are then explicitly constructed and the relations between them are found in the form of syzygies. In this way, complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.

A Bijection Between Paths for the {\mathcal{M}(p, 2p + 1)} Minimal Model Virasoro Characters

The states in the irreducible modules of the minimal models can be represented by infinite lattice paths arising from consideration of the corresponding RSOS statistical models. For the

Two-rowed Hecke algebra representations at roots of unity

{\mathcal{M}(p, 2p + 1)}

Construction of SO(5) SUPERSET OF SO(3) spherical harmonics and Clebsch-Gordan coefficients.

models, a completely different path representation has been found recently, this one on a half-integer lattice; it has no known underlying statistical-model interpretation. The correctness of this alternative representation has not yet been demonstrated, even at the level of the generating functions, since the resulting fermionic characters differ from the known ones. This gap is filled here, with the presentation of two versions of a bijection between the two path representations of the