Permutation Group Sn Research Articles

Clebsch-Gordan coefficients for the permutation group

A non-genealogical method is proposed for the reduction of the inner product representations of the permutation group SN. This method of determining the Clebsch-Gordan coefficients has been found to be recursive only within a given series. As such it permits the direct reduction of products of large-dimensional representations.

Particle statistics from induced representations of a local current group

Representations of the nonrelativistic current group 𝒮-𝒦 are studied in the Gel’fand–Vilenkin formalism, where 𝒮 is Schwartz’ space of rapidly decreasing functions, and 𝒦 is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of 𝒦F (the stability group of a point F∈𝒮′) which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a 𝒦 orbit Δ in 𝒮′, together with a suitable representation of 𝒦F for F∈Δ, sufficient conditions are developed for inducing a representation of 𝒮-𝒦. The Hilbert space for the induced representation consists of square-integrable functions on a covering space of Δ, which transform in accordance with a representation of 𝒦F. The Bose and Fermi N-particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of 𝒮-𝒦 determines a well-defined representation of 𝒦F; (2) equivalent representations of 𝒮-𝒦 determine equivalent representations of 𝒦F; (3) a representation of 𝒦F induces a representation of 𝒮-𝒦; and (4) equivalent representations of 𝒦F determine equivalent induced representations.

Isometries of diagonally symmetric Banach spaces

We describe all compact symmetric subgroups of the orthogonal groupO n which contain the permutation groupS n. There are seven such groups, each one can be realised as the group of all isometries on somen-dimensional Banach space.

A programmable spin-free method for configuration interaction

In this note a method is presented for quick implementation of configuration interaction (CI) calculations in molecules. A spin-free Hamiltonian for anN electron system in a spin stateS, expressed in terms of the generators for the unitary group algebra, is diagonalized over orbital configurations forming a basis for the irreducible representation [21/2N-S 12S ] of the permutation group S N . It has been found that the basic algebraic expressions necessary for the CI calculation involve a limited category of permutations. These have been displayed explicitly.

A method for the construction of orthogonal spin eigenfunctions

AbstractA method for the construction of the essentially idempotent and Hermitian diagonal elements of the matric algebra of the permutation group Sn is proposed. For the irreducible representation [λ] = [λ1, λ2] characterising a spin state S of an n‐electron system, it is found that this method generates the complete set of spin projections from the appropriate primitive spin functions. The method is applied to a 7‐electron system in the spin state S = MS = 1/2 and the results are listed in the Appendix.

On the Relation Between Isomerization Modes for Idealized and Distorted Skeleta

AbstractPermutational isomerization reactions[1][2] may occur in two extreme situations. If the n ligands do not distort the symmetry of the skeleton, then the permutations of the full permutation group Sn have to be classified [1,3] according to the symmetry group G of the idealized skeleton. If the ligands are sufficiently different, they will be blocked and will distort the skeletal framework. In this case, the allowed permutations belong to S′n, a subgroup of Sn and have to be classified [2] according to G′, the symmetry group of the distorted skeleton. In this paper, the relation between these two descriptions is analysed and illustrated on examples of chemical interest.

Antisymmetrization in relative co-ordinates. II: Fractional-parentage coefficients

In the framework of a theory to construct an antisymmetrized basis in relative co-ordinates for anN-nucleon system, a method to calculate the 〈N|N−1> fractional-parentage coefficients is given. The theory, which involves the main properties of the Young-Yamanouchi orthogonal representation of the permutation groupSN, gives the orbital f.p.c. as solutions of an algebraic system of linear homogeneous equations.

Characterization and Theoretical Comparison of Branch-and-Bound Algorithms for Permutation Problems

Branch-and-bound implicit enumeration algorithms for permutation problems (discrete optimization problems where the set of feasible solutions is the permutation group S n ) are characterized in terms of a sextuple ( B p S,E,D,L,U ), where (1) B p is the branching rule for permutation problems, (2) S is the next node selection rule, (3) E is the set of node elimination rules, (4) D is the node dominance function, (5) L is the node lower-bound cost function, and (6) U is an upper-bound solution cost. A general algorithm based on this characterization is presented and the dependence of the computational requirements on the choice of algorithm parameters, S, E, D, L, and U is investigated theoretically. The results verify some intuitive notions but disprove others.

Spin‐free wave functions in many‐electron perturbation theory. I. Closed‐shell systems

AbstractThe Schrödinger equation of an N‐electron closed‐shell system is reduced to that for the spatial wave function Ψ by the aid of the theory of the symmetric (permutation) group SN. The first‐order perturbation equation based on the Hartree–Fock‐SCF model as the zero‐order solution is then solved by decomposing it into a set of soluble 1/2(N/2)[(N/2) + 1] equations for the pair‐correlation (first‐order) spatial wave functions. These spatial functions give the spatial parts of the pair functions in the spin orbital basis formulation, reducing also the number of independent pairs needed in the first order.

Transformation Properties of Antisymmetric Spin Eigenfunctions under Linear Mixing of the Orbitals

After recalling the duality between the general linear group GL(m), represented by its N-fold inner product, and the permutation group SN, we have given a survey of its quantum chemical consequences. It causes the one-to-one correspondence between the total spin quantum number and the permutation symmetry of N-electron spin functions, and, via the Pauli principle which imposes permutation symmetry on the spatial part also, it leads to specific properties of antisymmetric spin eigenfunctions under orbital transformations. Such functions can be classified according to the irreducible representations of GL(m). For special orbital transformations, often occurring in quantum chemistry, which mix only orbitals in different subsets among each other, we have derived how the transformation of the N-electron wavefunctions simplifies, by a reduction of the representations of GL(m). The theory is illustrated by an example and some applications are discussed.