Tensor Product Decomposition Research Articles

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Isolate Domination Decomposition of Tensor Product of Cycle Related Graphs

Isolate Domination Decomposition of Tensor Product of Cycle Related Graphs

Quantum Tensor-Product Decomposition from Choi-State Tomography

The Schmidt decomposition is the go-to tool for measuring bipartite entanglement of pure quantum states. Similarly, it is possible to study the entangling features of a quantum operation using its operator-Schmidt or tensor-product decomposition. While quantum technological implementations of the former are thoroughly studied, entangling properties on the operator level are harder to extract in the quantum computational framework because of the exponential nature of sample complexity. Here, we present an algorithm for unbalanced partitions into a small subsystem and a large one (the environment) to compute the tensor-product decomposition of a unitary the effect of which on the small subsystem is captured in classical memory, while the effect on the environment is accessible as a quantum resource. This quantum algorithm may be used to make predictions about operator nonlocality and effective open quantum dynamics on a subsystem, as well as for finding low-rank approximations and low-depth compilations of quantum circuit unitaries. We demonstrate the method and its applications on a time-evolution unitary of an isotropic Heisenberg model in two dimensions. Published by the American Physical Society 2024

A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

A kite K is a graph which can be obtained by joining an edge to any vertex of K 3 . A kite with edge set { a b , b c , c a , c d } can be denoted as ( a , b , c ; c d ) . If every vertex of a kite in the decomposition lies in different partite sets, then we say that a kite decomposition of a multipartite graph is a gregarious kite decomposition. In this manuscript, it is shown that there exists a decomposition of ( K m ⊗ ¯¯¯¯¯ K n ) × ( K r ⊗ ¯¯¯¯¯ K s ) into gregarious kites if and only if n 2 s 2 m ( m − 1 ) r ( r − 1 ) ≡ 0 ( mod 8 ) , where ⊗ and × denote the wreath product and tensor product of graphs respectively. We denote a gregarious kite decomposition as G K -decomposition.

An adaptive support domain for the in-compressible fluid flow based on the localized radial basis function collocation method

In this paper, an adaptive support domain scheme is proposed to the in-compressible fluid flow based on the localized radial basis function collocation method (LRBFCM). The idea of the adaptive support domain avoids complex mathematical derivations of the finite difference method (FDM). Unlike other upwind schemes in the LRBFCM, the support domain uses the sampling nodes similar to different finite difference schemes, and the improved LRBFCM is further applied to the in-compressible fluid flows. Considering the Kronecker tensor product and Cholesky decomposition techniques, the improved LRBFCM can be easily used to the lid-driven problems with Reynolds numbers up to 100,000 by a simple mathematical derivation. The proposed LRBFCM has an adaptive support domain similar to the finite difference schemes with much less CPU time costs.

3j-symbols for tensor product of symmetric representations

We propose an algebraic procedure to obtain quantum 3j-symbols (quantum Clebsch–Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. In fact, these symbols will be useful to write the spectral parameter dependent R-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

Tensor product decomposed multi‐material isogeometric topology optimization with explicit NURBS element stiffness computation

AbstractBased on the explicit stiffness computation formula of NURBS (non‐uniform rational B‐splines) element and tensor product decomposed explicit filter, a novel multi‐material isogeometric topology optimization (MMITO) method is put forward to generate multi‐material structures efficiently. The explicit NURBS element stiffness computation formula is: derived from the Bėzier extraction operator under the combination of scaled and rotational geometric transformations between NURBS elements, and the stiffness matrices of NURBS elements can be computed by the stiffness matrix of a Bėzier element without resorting to the time‐consuming integral procedure. As the presented MMITO is a nested optimization framework, we apply the explicit filtering technique to update the multi‐phase design variables efficiently in combination with the tensor product decomposition method. With the decomposed weight matrices of the filter and the explicit form of the NURBS stiffness matrix, we obtain the equivalent but much simplified preprocessing procedure and sensitivity analysis for the MMITO method. A comprehensive set of multi‐material numerical examples are optimized to verify that, while improving the preprocessing efficiency and reducing the memory overhead, the present MMITO scheme is able to generate semblable two‐dimensional and three‐dimensional multi‐material structures with identical structural performance to the traditional scheme. Therefore, the proposed MMITO method is a promising approach to optimizing multi‐material structures.

On the multiplicity spaces of spherical subgroups of minimal rank

AbstractLet be a complex connected semisimple Lie group and be a connected semisimple subgroup of . Consider the branching problem of decomposing the simple ‐representations as a sum of simple ‐representations . When , it is the tensor product decomposition. The multiplicity space satisfies , where the sum runs over the isomorphism classes of simple ‐representations. In the case when is spherical of minimal rank, we describe as the intersection of kernels of powers of root operators in some weight space of the dual space of . When , we recover by geometric methods a well‐known result.

LEX-EFT: the Light Exotics Effective Field Theory

We propose the creation of a Light Exotics Effective Field Theory (LEX-EFT) catalog. LEX-EFT is a generic framework to capture all interactions between the Standard Model (SM) and all (or at least a large class of) theoretically allowed exotic states beyond the Standard Model (bSM), indexed by their SM and bSM charges. These states are light enough to be on or nearly on shell in some collider processes. This framework, which subsumes beyond the Standard Model paradigms as generally as possible, is meant to extend recent successful implementations of bSM EFTs and complement e.g. the Standard Model Effective Field Theory (SMEFT), which can capture the off-shell effects of exotic fields. In this work, we review a general method for the construction of a complete list of gauge-invariant operators involving SM interactions with light exotics via iterative tensor product decomposition, up to the desired order in mass dimension. Each operator is characterized by specific Clebsch-Gordan coefficients determined by the charge flow; we show how this charge flow affects the range of EFT validity and cross sections associated with an effective operator. We create an example catalog of exotic scalars coupling to SM gauge boson pairs, and we highlight some operators with exotic weak SU(2)L charges that can produce spectacular LHC phenomenology. We further demonstrate the utility of the LEX-EFT approach with several examples of effects on kinematic distributions and cross sections that would not be captured by EFTs agnostic to the exotic degrees of freedom and may evade the main inclusive collider searches tailored to the existing preferred set of standard bSM theories.

Local Unitary Equivalence of Quantum States Based on the Tensor Decompositions of Unitary Matrices.

Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.

Tensor product decomposition of crystal bases of type A by K-hives

The crystal basis of an irreducible highest weight module of type A is realized by a combinatorial model called K-hives. In this paper, we give the tensor product decomposition map of crystal bases of irreducible highest weight modules of type A in terms of K-hives. In the process of constructing the decomposition map, we provide a combinatorial description on K-hives of the decomposition map.

Product decompositions of moment-angle manifolds and B-rigidity

AbstractA simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal {Z}_P$ over P. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

Ghost center and representations of the diagonal reduction algebra of osp(1|2)

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra A of the orthosymplectic Lie superalgebra osp(1|2). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of A is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for osp(1|2)). As another application, we classify all finite-dimensional irreducible representations of A. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

Airy point process via supersymmetric lifts

We study the local asymptotics at the edge for particle systems arising from: (i) eigenvalues of sums of unitarily invariant random Hermitian matrices and (ii) signatures corresponding to decompositions of tensor products of representations of the unitary group. Our method treats these two models in parallel, and is based on new formulas for observables described in terms of a special family of lifts, which we call supersymmetric lifts, of Schur functions and multivariate Bessel functions. We obtain explicit expressions for a class of supersymmetric lifts inspired by determinantal formulas for supersymmetric Schur functions due to \cite{MJ03}. Asymptotic analysis of these lifts enable us to probe the edge. We focus on several settings where the Airy point process arises.

Congruence for Lattice Path Models with Filter Restrictions and Long Steps

We derive a path counting formula for a two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves the problem of finding an explicit formula for multiplicities of modules in tensor product decomposition of T(1)⊗N for Uq(sl2) with divided powers, where q is a root of unity. Combinatorial treatment of this problem calls for the definition of congruence of regions in lattice path models, properties of which are explored in this paper.

Some Unexpected Properties of Littlewood-Richardson Coefficients

We are interested in identities between Littlewood-Richardson coefficients, and hence in comparing different tensor product decompositions of the irreducible modules of the linear group $\text{GL}_n({\mathbb C})$. A family of partitions — called near-rectangular — is defined, and we prove a stability result which basically asserts that the decomposition of the tensor product of two representations associated to near-rectangular partitions does not depend on $n$. Given a partition $\lambda$, of length at most $n$, denote by $V_n(\lambda)$ the associated simple $\text{GL}_n({\mathbb C})$-module. We conjecture that, if $\lambda$ is near-rectangular and $\mu$ any partition, the decompositions of $V_n(\lambda)\otimes V_n(\mu)$ and $V_n(\lambda)^*\otimes V_n(\mu)$ coincide modulo a mysterious bijection. We prove this conjecture if $\mu$ is also near-rectangular and report several computer-assisted computations which reinforce our conjecture.

Nested fibre bundles in Bott-Samelson varieties

We consider Bott-Samelson varieties BSc(s) for a semisimple compact Lie group C corresponding to sequences of (not necessarily simple) reflections s. Let n be the length of s, K be a maximal torus in C and W be the Weyl group of C. For any set R of not overlapping integer pairs (i,j) such that 1⩽i⩽j⩽n and a function v:R→W, we consider the subspace BSc(s,v)⊂BSc(s) of solutions of the equations in C/K requiring that the K-orbit of the product of coordinates counted from i to j be equal to the K-orbit of v evaluated at (i,j)∈R.We decompose BSc(s,v) into a twisted product (in the sense of iterated fibre bundles) of smaller Bott-Samelson varieties BSc(t) and the fibres of the canonical projections from BSc(t) to the flag variety. Finally, we prove the tensor product decomposition for the K-equivariant cohomology of BSc(s,v).

On a ℂ2-valued integral index transform and bilateral hypergeometric series

We discuss the spectral decomposition of the hypergeometric differential operators on the line , such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group . Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.

Tensor product decompositions and rigidity of full factors

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g. $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.

Tensor Product and Tensor-Singular Value Decomposition Based Multi-Exposure Fusion of Images

Considering multidimensional structure of the multi-exposure images, a new Tensor product and Tensor-singular value decomposition based Multi-Exposure image Fusion (TT-MEF) method is proposed. The main innovation of this work is to explore a new feature representation of multi-exposure images in the new tensor domain and design the fusion strategy on this basis. Specifically, the luminance and the chrominance channels are fused separately to maintain color consistency. For the luminance fusion, the luminance channel of multi-exposure images is divided into two parts, that is, de-mean term and mean term. The de-mean term is represented as a tensor to extract the feature. Then, the tensor product and tensor-singular value decomposition (T-SVD) are used to design a tensor feature extractor. Furthermore, a fusion strategy of the de-mean term is presented according to the visual saliency model, and a fusion strategy of the mean term is defined by the local and the global visual weights to control counterpoise between the local and global luminance. For the chrominance fusion, a new fusion strategy is also designed by the tensor product and T-SVD, similar to the luminance fusion. Finally, the fused image is obtained by combining the luminance and chrominance fusion. Experimental results show that the proposed TT-MEF method generally outperforms the existing state-of-the-art in terms of subjective visual quality and objective evaluation.