Combinatorial aspects of representations of U(n)

The boson operator theory of the representations of the unitary group, its Wigner-Clebsch-Gordan, and Racah coefficients is reformulated in terms of the ring of polynomials in any number of indeterminates with the goal of bringing the theory, as nearly as possible, under the purview of combinatorial oriented concepts. Four of the basic relations in unitary group theory are interpreted from this viewpoint

Unitary symmetry, combinatorics, and special functions

From 1967 to 1994, Larry Biedenham and I collaborated on 35 papers on various aspects of the general unitary group, especially its unitary irreducible representations and Wigner-Clebsch-Gordan coefficients. In our studies to unveil comprehensible structures in this subject, we discovered several nice results in special functions and combinatorics. The more important of these will be presented and their present status reviewed

New relations and identities for generalized hypergeometric coefficients

AbstractGeneralized hypergeometric coefficients 〈pFq(a; b)¦λ〉 enter into the problem of constructing matrix elements of tensor operators in the unitary groups and are the expansion coefficients of a multivariable symmetric function generalization pFq(a; b; z), z = (z1, z2,…, zt), of the Gauss hypergeometric function in terms of the Schur functions eλ(z), where λ = (λ1, λ2,…, λt) is an arbitrary partition. As befits their group-theoretic origin, identities for these generalized hypergeometric coefficients characteristically involve series summed over the Littlewood-Richardson numbers g(μνλ). Identities that may be interpreted as generalizations of the Bailey, Saalschütz,… identities are developed in this paper. Of particular interest is an identity which develops in a natural way a group-theoretically defined expansion over new inhomogeneous symmetric functions. While the relations obtained here are essential for the development of the properties of tensor operators, they are also of interest from the viewpoint of special functions

An alternative approach to the construction of Schur-Weyl transform

We propose an alternative approach for the construction of the unitary matrix which performs generalized unitary rotations of the system consisting of independent identical subsystems (for example spin system). This matrix, when applied to the system, results in a change of degrees of freedom, uncovering the information hidden in non-local degrees of freedom. This information can be used, inter alia, to study the structure of entangled states, their classification and may be useful for construction of quantum algorithms.Comment: 6 page

Survey of zeros of 3j and 6j coefficients by Diophantine equation methods

The general polynomials whose positive integer solutions determine all nontrivial zeros of 3j and 6j coefficients are given. The determination of weight-1 and weight-2 zeros from classical Diophantine equations is reviewed. 25 refs

The SU(3) generalizatoin of Racah's SU(2j + 1) contains SU(2) group-subgroup embedding

Racah showed how to embed the symmetry group, SU(2) or SO(3), of a physical system in the general unitary group SU(2j + 1), where the latter group is the most general group of linear transformations of determinant 1 that leaves invariant the inner product structure of an arbitrary state space H{sub j} of the physical system. This state space H{sub j} is at the same time the carrier space of an irreducible representation (irrep) (j) of the symmetry group. This embedding is achieved by classifying the vector space of mappings H{sub j} {yields} H{sub j} as irreducible tensor operators with respect to the underlying symmetry group. These irreducible tensors are the generators of the Lie algebra of SU(2j + 1). Racah's method is reviewed within the framework of unit tensor operators. The generalization of this technique to the symmetry group U(3) to obtain the embedding U(3) {contained in} U(n), where n = dim(m) is the dimension of an arbitrary irrep of U(3). As in the SU(2) case, the group U(3) is the symmetry group of a physical system, and U(dim(m)) is the most general group of linear transformations that preserves the inner product structure of an arbitrary state space H{sub (m)} of the system. This state space H{sub (m)} is at the same time the carrier space of irrep (m) of the symmetry group U(3). Preliminary results on the Lie algebraic vanishings of U(3) Racah coefficients in consequence of the embedding SU(3) {contained in} E{sub 6} {contained in} SU(27) are given. 61 refs

Representations of the symmetric group as special cases of the boson polynomials in U(n)

The set of all real, orthogonal irreps of S/sub n/ are realized explicitly and nonrecursively by specializing the boson polynomials carrying irreps of the unitary group. This realization makes use of a 'calculus of patterns', which is discussed

Combinatorics, geometry, and mathematical physics

This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). Combinatorics and geometry have been among the most active areas of mathematics over the past few years because of newly discovered inter-relations between them and their potential for applications. In this project, the authors set out to identify problems in physics, chemistry, and biology where these methods could impact significantly. In particular, the experience suggested that the areas of unitary symmetry and discrete dynamical systems could be brought more strongly under the purview of combinatorial methods. Unitary symmetry deals with the detailed description of the quantum mechanics of many-particle systems, and discrete dynamical systems with chaotic systems. The depth and complexity of the mathematics in these physical areas of research suggested that not only could significant advances be made in these areas, but also that here would be a fertile feedback of concept and structure to enrich combinatorics itself by setting new directions. During the three years of this project, the goals have been realized beyond expectation, and in this report the authors set forth these advancements and justify their optimism

Identities for generalized hypergeometric coefficients

Generalizations of hypergeometric functions to arbitrarily many symmetric variables are discussed, along with their associated hypergeometric coefficients, and the setting within which these generalizations arose. Identities generalizing the Euler identity for {sub 2}F{sub 1}, the Saalschuetz identity, and two generalizations of the {sub 4}F{sub 3} Bailey identity, among others, are given. 16 refs