Tensor Product Decomposition Research Articles

The morphology of 𝒩 = 6 Chern-Simons theory

We tabulate various properties of the language of N=6 Chern-Simons Theory, in the sense of Polyakov. Specifically we enumerate and compute character formulas for all syllables of up to four letters, i.e. all irreducible representations of OSp(6|4) built from up to four fundamental fields of the ABJM theory. We also present all tensor product decompositions for up to four singletons and list the (cyclically invariant) four-letter words, which correspond to single-trace operators of length four. As an application of these results we use the two-loop dilatation operator to compute the leading correction to the Hagedorn temperature of the weakly-coupled planar ABJM theory on R \times S^2.

A novel realization of the Virasoro algebra in number state space

The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d ⩾ 2 , accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for s u ( 1 , 1 ) . As a step towards the solution, a realization of the ( d = 1 ) Virasoro algebra Vir ∼ Diff + ( S ( 1 ) ) in the universal envelope of su ( 1 , 1 ) (and h ( 1 ) ) is presented, which is simple in the discrete positive series irreducible unitary representation D κ ( + ) of s u ( 1 , 1 ) .

Tensor product decompositions for su(3) of an irreducible representation with itself and with its conjugate

In this article a method to obtain the decomposition, including multiplicities, of an su(3) irreducible representation with itself and with its conjugate, is provided. The decomposition of such tensor products plays an important role in physics. The objective is to provide a simple method for performing these decompositions in addition to showing how this method can be used to derive several interesting facts about the decompositions, particularly in regards to multiplicities.

Representations of quantum permutation algebras

We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π : A s ( n ) → B ( H ) . We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n = 6 .

Diagonal free field matrix correlators, global symmetries and giant gravitons

We obtain a basis of diagonal free field multi-matrix 2-point correlators in a theory with global symmetry group G. The operators fall into irreducible representations of G. This applies for gauge group U(N) at finite N. For composites made of n fundamental fields, this is expressed in terms of Clebsch-Gordan coefficients for the decomposition of the n-fold tensor products of the fundamental field representation in terms of G \times S_n representations. We use this general construction in the case of the SL(2) sector of \cN=4 SYM. In this case, by using oscillator constructions, we reduce the computation of the relevant Clebsch-Gordans coupling infinite dimensional discrete series irreps of SL(2) to a problem in symmetric groups. Applying these constructions we write down gauge invariant operators with a Fock space structure similar to that arising in a large angular momentum limit of worldvolume excitations of giant gravitons. The Fock space structure emerges from Clebsch multiplicities of tensor products of symmetric group representations. We also give the action of the 1-loop dilatation operator of \cN=4 SYM on this basis of multi-matrix operators.

The Planar Rook Algebra and Pascal's Triangle

. Even themultiplicative structure of the binomial coecients arises from the representa-tion theory of the planar rook algebras, as we discover upon decomposition oftensor product representations.The planar rook algebra is an example of a \diagram algebra, which forour purposes is a nite-dimensional algebra with a basis given by a collectionof diagrams and multiplication described combinatorially by diagram concate-nation. When the basis diagrams can be drawn without edge crossings, we geta planar algebra. There is a growing theory of planar algebras initiated by V.Jones (see [Jo]) that uses a more re ned de nition of planarity than what wegive here.The main goal of this paper is to work out the combinatorial representationtheory of the planar rook algebra CP

The Eigencone and Saturation for Spin(8)

We explicitly calculate the system of restricted triangle inequalities for the group PSO(8) given by Belkale-Kumar, thereby explicitly solving the eigenvalues of a sum problem for this group (equivalently describing the side-lengths of geodesic triangles in the corresponding symmetric space for the metric d∆ with values in the Weyl chamber ∆). We then apply some computer programs to verify the saturation conjecture for the decomposition of tensor products of finite-dimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights (λ, μ, ν) such that λ + μ + ν is in the root lattice, and any positive integer N , (V (λ)⊗ V (μ)⊗ V (ν)) 6= 0 if and only if (V (Nλ)⊗ V (Nμ)⊗ V (Nν)) 6= 0.

A max-flow algorithm for positivity of Littlewood-Richardson coefficients

Littlewood-Richardson coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group $\mathrm{GL}(n,\mathbb{C})$. They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit $\textit{combinatorial}$ polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks. Les coefficients de Littlewood-Richardson sont les multiplicités dans la décomposition du produit tensoriel de deux représentations irréductibles du groupe général linéaire $\mathrm{GL}(n,\mathbb{C})$. Ces coefficients ont plusieurs interprétations en combinatoire, en théorie des représentations et en géométrie. Mulmuley et Sohoni ont observé qu'on peut décider si un coefficient de Littlewood-Richardson est positif en temps polynomial. C'est une conséquence de la propriété de saturation des coefficients de Littlewood-Richardson (démontrée par Knutson et Tao en 1999) et le fait bien connu que la programmation linéaire est possible en temps polynomial. Nous décrivons un algorithme $\textit{combinatoire}$ pour décider si un coefficient de Littlewood-Richardson est positif. Cet algorithme est bien adapté au problème et il utilise des idées de la théorie des flots maximaux sur des réseaux.

Sl(1|2) Super-Toda Fields

Explicit exact solution of supersymmetric Toda fields associated with the Lie superalgebra sl(2|1) is constructed. The approach used is a super extension of Leznov–Saveliev algebraic analysis, which is based on a pair of chiral and antichiral Drienfeld–Sokolov systems. Though such approach is well understood for Toda field theories associated with ordinary Lie algebras, its super analogue was only successful in the super Liouville case with the underlying Lie superalgebra osp(1|2). The problem lies in that a key step in the construction makes use of the tensor product decomposition of the highest weight representations of the underlying Lie superalgebra, which is not clear until recently. So our construction made in this paper presents a first explicit example of Leznov–Saveliev analysis for super Toda systems associated with underlying Lie superalgebras of the rank higher than 1.

Some remarks on the ring of twisted tilting modules for algebraic groups

In a recent paper [S. Doty, A. Henke, Decomposition of tensor products of modular irreducibles for SL 2, Q. J. Math. 56 (2005) 189–207], Doty and Henke give a decomposition of the tensor product of two rational simple modules for the special linear group of degree 2 over an algebraically closed field of characteristic p > 0 . In performing this calculation it proved useful to know that the simple modules are twisted tensor products of tilting modules. It seems natural therefore to consider the ring of twisted tilting modules for a semisimple group G (a subring of the representation ring of G). However, we quickly specialize to the case in which G is the special linear group of degree 2. We show that (in this case) the ring is reduced and describe associated varieties. We give formulas from which one may determine the multiplicities of the indecomposable module summands of the tensor product of twisted tilting modules.

A geometric description of tensor product decompositions in su(3)

The direct sum decomposition of tensor products for su(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary and helping draw general conclusions. After a description and proof of the method, several consequences are discussed and proved. In particular, questions regarding multiplicities are considered.

Hopf algebras and characters of classical groups

Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of n-independent, universal characters. The corresponding rings, CharGL, CharO and CharSp, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case.

Representations of Nonstandard Poincaré Hopf Algebras

Let 𝔭 q (1 + 1) be a nonstandard Poincaré Hopf algebra, we characterize all finite dimensional completely E-semisimple modules of 𝔭 q (1 + 1). We also classify all finite dimensional E-semisimple modules of 𝔭 q for a special quotient algebra of 𝔭 q (1 + 1). Moreover, the decomposition of tensor product of two finite dimensional E-semisimple indecomposable modules is obtained.

Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

We analyze the structure of the algebra $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. In the case $|\mathbf{x}|= \infty$, our techniques simplify to a form readily generalized to many other familiar pairs of combinatorial Hopf algebras. Nous analysons la structure de l'algèbre $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ des polynômes symétriques en des variables non-commutatives pour obtenir des analogues des résultats classiques concernant la structure de l'anneau $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ des polynômes symétriques en des variables commutatives. Plus précisément, au moyen de "l'action par positions'', on réalise $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ comme sous-module de $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$. On découvre alors une nouvelle décomposition de $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ comme produit tensoriel, obtenant ainsi un analogue des théorèmes classiques de Chevalley et Shephard-Todd. Dans le cas $|\mathbf{x}|= \infty$, nos techniques se simplifient en une forme aisément généralisables à beaucoup d'autres paires d'algèbres de Hopf familières.

The complexity of computing Kronecker coefficients

Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group $S_n$. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem $\mathrm{KRONCOEFF}$ of computing Kronecker coefficients is very difficult. More specifically, we prove that $\mathrm{KRONCOEFF}$ is #$\mathrm{P}$-hard and contained in the complexity class $\mathrm{GapP}$. Formally, this means that the existence of a polynomial time algorithm for $\mathrm{KRONCOEFF}$ is equivalent to the existence of a polynomial time algorithm for evaluating permanents. Les coefficients de Kronecker sont les multiplicités dans la décomposition du produit tensoriel de deux représentations irréductibles du groupe symétrique. On peut aussi les voir comme les coefficients du développement du produit interne des polynômes de Schur. Nous montrons que le problème $\mathrm{KRONCOEFF}$ de calculer les coefficients de Kronecker est très difficile. Plus précisément, nous prouvons que $\mathrm{KRONCOEFF}$ est #$\mathrm{P}$-dur et que $\mathrm{KRONCOEFF}$ est dans la classe de complexité $\mathrm{GapP}$. Cela veut dire qu'il existe un algorithme pour $\mathrm{KRONCOEFF}$ s'exécutant en temps polynomial si et seulement s'il existe un algorithme pour l'évaluation du permanent s'exécutant en temps polynomial.

The PRV-formula for tensor product decompositions and its applications

Let G be a semisimple algebraic group, V a simple finite-dimensional self-dual G-module, and W an arbitrary simple finite-dimensional G-module. Using the triple multiplicity formula due to Parthasarathy, Ranga Rao, and Varadarajan, we describe the multiplicities of W in the symmetric and exterior squares of V in terms of the action of a maximum-length element of the Weyl group on some subspace in V T , where T ⊂ G is a maximal torus. By way of application, we consider the cases in which V is the adjoint, little adjoint, or, more generally, a small G-module. We also obtain a general upper bound for triple multiplicities in terms of Kostant’s partition function.

Finite-dimensional representations of hyper loop algebras

We study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over an algebraically closed field of positive characteristic associated to the loop algebra over a complex finite-dimensional simple Lie algebra. The main results are the classification of the irreducible modules, a version of Steinberg's Tensor Product Theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero setting. Furthermore, we start the study of reduction modulo p and prove that every irreducible module of a hyper loop algebra can be constructed as a quotient of a module obtained by a certain reduction modulo p process applied to a suitable characteristic zero module. We conjecture that the Weyl modules are also obtained by reduction modulo p. The conjecture implies a tensor product decomposition for the Weyl modules which we use to describe the blocks of the underlying abelian category.

Optimal blended spectral-element operators for acoustic wave modeling

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D factors. We apply standard lumping to the factor associated with the 1D mass matrix. A simplified numerical dispersion analysis of arbitrary order and spatial dimension provides a practical criterion for weighting consistent and lumped matrices. The approach is general and is valid for solving both the time-dependent and the stationary (Helmholtz) wave equations.

Tensor product approximation with optimal rank in quantum chemistry

Tensor product decompositions with optimal separation rank provide an interesting alternative to traditional Gaussian-type basis functions in electronic structure calculations. We discuss various applications for a new compression algorithm, based on the Newton method, which provides for a given tensor the optimal tensor product or so-called best separable approximation for fixed Kronecker rank. In combination with a stable quadrature scheme for the Coulomb interaction, tensor product formats enable an efficient evaluation of Coulomb integrals. This is demonstrated by means of best separable approximations for the electron density and Hartree potential of small molecules, where individual components of the tensor product can be efficiently represented in a wavelet basis. We present a fairly detailed numerical analysis, which provides the basis for further improvements of this novel approach. Our results suggest a broad range of applications within density fitting schemes, which have been recently successfully applied in quantum chemistry.

Bogoliubov Transformations and Entanglement of Two Fermions

We show that Bogoliubov transformations, widely used in quantum field theory, can be also useful in quantum information theory. Namely, we show that the problem of the choice of tensor product decomposition in the system of two fermions can be analysed with the help of Bogoliubov transformations of creation and annihilation operators.