Tensor Product Decomposition Research Articles

Generating functions for tensor product decomposition

The paper deals with the tensor product decomposition problem. Tensor product decompositions are of great importance in the quantum physics. A short outline of the state of the art for the of semisimple Lie groups is mentioned. The generality of generating functions is used to solve tensor products. The corresponding generating function is rational. The feature of this technique lies in the fact that the decompositions of all tensor products of all irreducible representations are solved simultaneously. Obtaining the generating function is a difficult task in general. We propose some changes to an algorithm using Patera-Sharp character generators to find this generating function, which simplifies the whole problem to simple operations over rational functions.

Real Dimension Groups

AbstractDimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Unique prime decomposition results for factors coming from wreath product groups

We use malleable deformations combined with spectral gap rigidity theory, in the framework of Popa's deformation/rigidity theory to prove unique tensor product decomposition results for II$_1$ factors arising as tensor product of wreath product factors. We also obtain a similar result regarding measure equivalence decomposition of direct products of such groups.

VON NEUMANN ENTROPY AND RELATIVE POSITION BETWEEN SUBALGEBRAS

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

Models for modules: the story of

We recall the structure of the indecomposable sl(2) modules in the Bernstein–Gelfand–Gelfand category . We show that all these modules can arise as quantized phase spaces of physical models. In particular, we demonstrate in a path integral discretization how a redefined action of the sl(2) algebra over the complex numbers can glue finite dimensional and infinite dimensional highest weight representations into indecomposable wholes. Furthermore, we discuss how projective cover representations arise in the tensor product of finite dimensional and Verma modules and give explicit tensor product decomposition rules. The tensor product spaces can be realized in terms of product path integrals. Finally, we discuss relations of our results to brane quantization and cohomological calculations in string theory.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

In this paper we present Affine.m—a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Program summaryProgram title: Affine.mCatalogue identifier: AENA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, UKLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 24844No. of bytes in distributed program, including test data, etc.: 1045908Distribution format: tar.gzProgramming language: Mathematica.Computer: i386–i686, x86_64.Operating system: Linux, Windows, Mac OS, Solaris.RAM: 5–500 MbClassification: 4.2, 5.Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems.Solution method:We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases.Restrictions:Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical.Unusual features:We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface.Running time:From seconds to days depending on the rank of the algebra and the complexity of the representation.

A combinatorial decomposition of higher level Fock spaces

We give a simple characterization of the highest weight vertices in the crystal graph of the level l Fock spaces. This characterization is based on the notion of totally periodic symbols viewed as affine analogues of reverse lattice words classically used in the decomposition of tensor products of fundamental $\mathfrak{sl}_{n}$-modules. This yields a combinatorial decomposition of the Fock spaces in their irreducible components and the branching law for the restriction of the irreducible highest weight $\mathfrak{sl}_{\infty}$-modules to $\widehat{\mathfrak{sl}_{e}}$.

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in $${\mathbb{P}^3}$$

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in \({\mathbb{P}^3}\) does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.

Matrix-Valued Orthogonal Polynomials Related to (SU(2)×SU(2), diag)

The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a matrix-valued function, which are the full spherical functions. Their orthogonality is a consequence of the Schur orthogonality relations. From the full spherical functions we obtain matrix-valued orthogonal polynomials of arbitrary size, and they satisfy a three-term recurrence relation which follows by considering tensor product decompositions. An explicit expression for the weight and the complete block-diagonalization of the matrix-valued orthogonal polynomials is obtained. From the explicit expression we obtain right-hand sided differential operators of first and second order for which the matrix-valued orthogonal polynomials are eigenfunctions. We study the low-dimensional cases explicitly, and for these cases additional results, such as the Rodrigues formula and being eigenfunctions to first order differential-difference and second order differential operators, are obtained.

An inequality of Kostka numbers and Galois groups of Schubert problems

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of $\mathfrak{sl}_2\mathbb{C}$ -modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral. On montre que le groupe de Galois de tout problème de Schubert concernant des droites dans l'espace projective contient le groupe alterné. En utilisant un critère de Vakil et l'argument de position spéciale due à Schubert, ce résultat se déduit d'une inégalité particulière des nombres de Kostka des tableaux ayant deux rangées. Dans la plupart des cas, une injection combinatoriale facile montre l’inégalité. Pour les cas restants, on utilise le fait que ces nombres de Kostka apparaissent dans la décomposition en produit tensoriel des $\mathfrak{sl}_2\mathbb{C}$-modules. En interprétant le produit tensoriel comme l'action de certaines matrices de Toeplitz commutant entre elles, et en utilisant de l'analyse spectrale et les séries de Fourier, on réécrit l’inégalité comme la positivité d'une intégrale. L’inégalité sera établie en estimant cette intégrale.

Preservation of Ranking Order in the Expression of Human Housekeeping Genes

Housekeeping (HK) genes fulfill the basic needs for a cell to survive and function properly. Their ubiquitous expression, originally thought to be constant, can vary from tissue to tissue, but this variation remains largely uncharacterized and it could not be explained by previously identified properties of HK genes such as short gene length and high GC content. By analyzing microarray expression data for human genes, we uncovered a previously unnoted characteristic of HK gene expression, namely that the ranking order of their expression levels tends to be preserved from one tissue to another. Further analysis by tensor product decomposition and pathway stratification identified three main factors of the observed ranking preservation, namely that, compared to those of non-HK (NHK) genes, the expression levels of HK genes show a greater degree of dispersion (less overlap), stableness (a smaller variation in expression between tissues), and correlation of expression. Our results shed light on regulatory mechanisms of HK gene expression that are probably different for different HK genes or pathways, but are consistent and coordinated in different tissues.

CONVERGENCE OF A GREEDY ALGORITHM FOR HIGH-DIMENSIONAL CONVEX NONLINEAR PROBLEMS

In this paper, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.

Generators for decompositions of tensor products of modules

If K is a field of finite characteristic p, G is a cyclic group of order q = pα, U and W are indecomposable KG-modules, and p ≥ dim U + dim W − 1, we describe how to find a generator for each of the indecomposable components of the KG-module \({U \otimes W}\).

Rankin-Cohen Brackets for Jacobi Forms and Tensor Products of Discrete Series Representations of the Jacobi Group

Explicit formulae for Rankin–Cohen brackets for Jacobi forms are obtained by pure Lie theoretic means. The Lie algebraic nature of these holomorphic bi-differential operators, which map pairs of Jacobi forms onto another such form, is unveiled, along with their relationship with the decompositions of tensor products of two (anti-)holomorphic discrete series representations of the Jacobi group into direct sums of similar representations. These decompositions, which turn out to be nonmultiplicity-free, are a problem of an independent interest, which is explicitly solved. Some of the obtained brackets are shown to be intertwiners, projecting the appropriate tensor product onto a direct summand. The first mentioned goal is reached by characterizing all the -equivariant holomorphic bi-differential operators, through establishing a one-to-one correspondence between them and the null vectors in the tensor products of two Verma -modules, for the complexified Lie algebra of . A large class of pairs of such modules is considered, the remaining situations being relegated to a future work. A thorough analysis of the structure of the tensor products, based on Borho’s isomorphism, which provides a factorization of the universal enveloping algebra of , yields explicit formulae for the sought after null vectors; they turn out to be completely determined by the null vectors in tensor products of three Verma -modules. The bi-differential operators associated with the obtained null vectors are then shown to be Rankin–Cohen brackets for Jacobi forms. The Lie theoretic approach developed here supplies a unifying framework within which results of previous investigations undertaken by different authors, using techniques pertaining to the realm of number theory, are recovered, provided with new interpretations and generalized. In particular, Bocherer’s generic dimensions are shown to be given by the above-mentioned multiplicities.

Structure Constants of Diagonal Reduction Algebras of gl Type

We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra $\gl_n$ into $\gl_n\oplus\gl_n$. Its representation theory is related to the theory of decompositions of tensor products of $\gl_n$-modules.

CleGo: A package for automated computation of Clebsch–Gordan coefficients in tensor product representations for Lie algebras [formula omitted

We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A–G based on the explicit construction of weight states. This straightforward approach (which is slower and more memory-consumptive than the standard methods to just calculate dimensions of the tensor product decomposition) produces Clebsch–Gordan coefficients that are of interest for instance in discussing symmetry breaking in model building for grand unified theories. For that purpose, multiple tensor products have been implemented as well as means for analyzing the resulting effective operators in particle physics. Program summaryProgram title: CleGoCatalogue identifier: AEIQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3641No. of bytes in distributed program, including test data, etc.: 34 536Distribution format: tar.gzProgramming language: OʼCamlComputer: i386–i686, x86_64Operating system: Cross-platform, for definiteness though we assume some UNIX system.RAM: ⩾4 GB commendable, though in general memory requirements depend on the size of the Lie algebras and the representations involved.Classification: 4.2, 11.1Nature of problem: Clebsch–Gordan coefficients are widely used in physics. This program has been written as a means to analyze symmetry breaking in the context of grand unified theories in particle physics. As an example, we computed the singlets appearing in higher-dimensional operators 27⊗27⊗27⊗78 and 27⊗27⊗27⊗650 in an E6-symmetric GUT.Solution method: In contrast to very efficient algorithms that also produce tensor product decompositions (as far as outer multiplicities/Littlewood–Richardson coefficients are concerned) we proceed straightforwardly by constructing all the weight states, i.e. the Clebsch–Gordan coefficients. This obviously comes at the expense of high memory and CPU-time demands. Applying Dynkin arithmetic in weight space, the algorithm is an extension of the one for the addition of angular momenta in su(2)≈A1, for reference see [1]. Note that, in general, Clebsch–Gordan coefficients are basis-dependent and therefore need to be understood with respect to the chosen basis. However, singlets appearing in (multiple) tensor products are less basis-dependent.Restrictions: Generically, only tensor products of non-degenerate or adjoint representations can be computed. However, the irreps appearing therein can subsequently be used as new input irreps for further tensor product decomposition so in principle there is no restriction on the irreps the tensor product is taken of. In practice, though, it is by the very nature of the explicit algorithm that input is restricted by memory and CPU runtime requirements.Unusual features: Analytic computation instead of float numerics.Additional comments: The program can be used in “notebook style” using a suitable OʼCaml toplevel. Alternatively, an OʼCaml input file can be compiled which results in processing that is approximately a factor of five faster. The latter mode is commendable when large irreps need to be constructed.Running time: Varies depending on the input from parts of seconds to weeks for very large representations (because of memory exhaustion).Reference:[1]J. Fuchs, C. Schweigert, Symmetries, Lie Algebras and Representations, Cambridge Univ. Press, Cambridge, UK, 1997, 438 p.

Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to [formula omitted

We study the tensor structure of the category of finite-dimensional modules of the restricted quantum enveloping algebra associated to sl 2 . Tensor product decomposition rules for all indecomposable modules are explicitly given. As a by-product, it is also shown that the category of finite-dimensional modules of the restricted quantum enveloping algebra associated to sl 2 is not a braided tensor category.

Husimi coordinates of multipartite separable states

A parametrization of multipartite separable states in a finite-dimensional Hilbert space is suggested. It is proved to be a diffeomorphism between the set of zero-trace operators and the interior of the set of separable density operators. The result is applicable to any tensor product decomposition of the state space. An analytical criterion for separability of density operators is established in terms of the boundedness of a sequence of operators.

Bethe ansatz solution of an integrable, non-Abelian anyon chain with [formula omitted] symmetry

The exact solution for the energy spectrum of a one-dimensional Hamiltonian with local two-site interactions and periodic boundary conditions is determined. The two-site Hamiltonians commute with the symmetry algebra given by the Drinfeld double D ( D 3 ) of the dihedral group D 3 . As such the model describes local interactions between non-Abelian anyons, with fusion rules given by the tensor product decompositions of the irreducible representations of D ( D 3 ) . The Bethe ansatz equations which characterise the exact solution are found through the use of functional relations satisfied by a set of mutually commuting transfer matrices.

On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

It is known that any quantum algorithm for graph isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across $\Omega(n\log n)$ coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This “quantum sieve” starts with coset states and works its way down toward representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for graph isomorphism. Specifically, we consider the natural reduction of graph isomorphism to the HSP over the wreath product $S_n\wr\mathbb{Z}_2$. Using a recently proved bound on the irreducible characters of $S_n$, we show that no algorithm in this family can solve graph isomorphism in less than $\mathrm{e}^{\Omega(\sqrt{n})}$ time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time $\mathrm{e}^{O(\sqrt{n\log n})}$.