Tuong Ton-That

n-Fold tensor products of GL(N, C) and decomposition of Fock spaces

Abstract Fock spaces over n × N complex matrices are introduced to deal with the reduction of n-fold tensor products of representations of GL(N, C ). These spaces are decomposed into orthogonal direct sums of finite-dimensional polynomial spaces P (M) that are invariant under GL(N, C ). A Frobenius reciprocity theorem is proved which relates the number of times an irreducible representation of GL(N, C ) appears in P (M) to the number of times this irreducible representation contains a representation of its diagonal subgroup. A GL(n, C ) action is shown to commute with the GL(N, C ) action and is used to find maps that send irreducible representations of GL(N, C ) into the P (M) spaces. Invariant Casimir operators are introduced to deal with the multiplicity occurring in n-fold tensor products.

Calculation of Clebsch-Gordan and Racah coefficients using symbolic manipulatio n programs

A procedure for calculating Clebsch--Gordan and Racah coefficients arising from the decomposition of n-fold tensor products of the U(N) groups using symbolic manipulation programs is given. Basic states are realized as polynomials in a space equipped with a differentiation inner product. The desired coefficients are then obtained by differentiating the relevant polynomials. Examples from SU(2) and SU(3) are given. copyright 1989 Academic Press, Inc.

On resolving the multiplicity of arbitrary tensor products of the U(N) groups

Representations of U(N) are realised as right translations on holomorphic Hilbert (Bargmann) spaces of n*N complex variables. r-fold tensor product spaces of irreducible representations of U(N) are shown to be isomorphic to subspaces of the holomorphic Hilbert spaces. Maps are exhibited which carry an irreducible representation of U(N) into these subspaces. The algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity. Several examples from U(3) are explicitly worked out.

Invariant theory of the block diagonal subgroups of GL(n,C) and generalized Casimir operators

Abstract Let C be the field of complex numbers and let GL ( n , C ) denote the general linear group of order n over C . Let n be the sum of m positive integers p 1 , …, p m and consider the block diagonal subgroup GL ( p 1 , C )x…x GL ( p m , C ). The adjoint representation of the Lie group GL ( n , C ) on its Lie algebra gives rise to the coadjoint representation of GL ( n , C ) on the symmetric algebra of all polynomial functions on GL ( n , C ). The polynomials that are fixed by the restriction of the coadjoint representation to the block diagonal subgroup form a subalgebra called the algebra of invariants. A finite set of generators of this algebra is explicitly determined and the connection with the generalized Casimir invariant differential operators is established.

Representations of Sn*U(N) in repeated tensor products of the unitary groups

The n-fold tensor product space generated by a given irreducible representation of the unitary group U(N) is a representation space for the symmetric group Sn as well as for U(N). Using ideas from the theory of dual pairs, such tensor product spaces are decomposed into irreducible representations of U(N) times reducible representations of Sn. Computationally effective formulae for the multiplicity of irreducible representations of Sn are given. Generating sets of invariant polynomials from the enveloping algebra of U(N) that commute with the Sn action are exhibited; it is shown that the eigenvalues of such operators can be used to break the multiplicity occurring in the dual U(N)xSn action.

Invariant theory, generalized Casimir operators, and tensor product decompositions of U(N)

One of the central problems in the representation theory of compact groups concerns multiplicity, wherein an irreducible representation occurs more than once in the decomposition of the n-fold tensor product of irreducible representations. The problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition. In this paper we use invariant theory along with so-called generalized Casimir operators to show how to resolve the multiplicity problem for the U(N) groups. The starting point is to augment the n-fold tensor product space with the contragredient representation of interest and construct a subspace of U(N) invariants. The setting for this construction is a polynomial space embedded in a Fock space of complex variables which carries all the irreducible representations of U(N) (or GLN ). The dimension of the invariant subspace is equal to the multiplicity occurring in the tensor product decomposition. Generalized Casimir operators are operators from the universal enveloping algebra of outer product U(N) groups that commute with the diagonal U(N) action and whose eigenvalues can be used to label the multiplicity. Using the notion of dual representations we show how to rewrite the generalized Casimir operators and prove that they act invariantly on the invariant subspace. A complete set of commuting generalized Casimir operators can therefore be used to construct eigenvectors that form an orthonormal basis in the invariant subspace. Different sets of generalized commuting Casimir operators generate different orthonormal bases in the invariant subspace; the overlaps between the eigenvectors of different commuting sets of generalized Casimir operators are called invariant coefficients. We show that Racah coefficients are special cases of invariant coefficients in which the generalized Casimir operators have been chosen with respect to a definite coupling scheme in the tensor product. The paper concludes with an example of the threefold tensor product of the eight-dimensional irreducible representation of U(3) in which the multiplicity of the chosen irreducible representation is 6. Eigenvectors in the six-dimensional invariant subspace are computed for different sets of generalized Casimir operators and invariant coefficients, including Racah coefficients.

Multiplicity, invariants and tensor product decompositions of tame representations of U(∞)

The structure of r-fold tensor products of irreducible tame representations of U(∞)=lim lim → U(n) are described, versions of contragredient representations and invariants are realized, and methods of calculating multiplicities, Clebsch–Gordan, and Racah coefficients are given using invariant theory on Bargmann–Segal–Fock spaces.

Multiplicity, invariants, and tensor product decompositions of compact groups

Decomposing tensor products of irreducible representations of compact groups almost always involves multiplicity, wherein some irreducible representations occur more than once in the direct sum decomposition. We show that the multiplicity can always be specified by polynomial group invariants. The setting is a Bargmann–Segal–Fock space in n×N complex variables, where n is the number of labels needed to specify the tensor product and N is the dimension of the fundamental representation of the compact group. Both the tensor product and direct sum bases are realized as polynomials in this space, and it is shown how Clebsch–Gordan and Racah coefficients can be computed by suitably differentiating these polynomials. The example of SU(N) is discussed in detail, and it is shown that the multiplicity can be computed as the solution of certain diophantine equations arising from powers of group invariants, namely minors of determinants.

CASIMIR OPERATORS OF SEMIDIRECT PRODUCTS OF SEMISIMPLE WITH HEISENBERG GROUPS

Casimir operators for semidirect products of some semisimple groups with Heisenberg groups are computed. The analysis is carried out using dual representations on Fock space, wherein the action of the semidirect products are related to their dual groups, namely certain unitary, orthogonal, and symplectic groups. The compact symplectic group chain is also investigated; by passing to the complexification, groups `between` the symplectic groups are constructed, which are of the form of semidirect products of symplectic groups with Heisenberg groups.

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyls branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples.