Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

关于矩阵张量积的一类问题

本文给出有限个矩阵张量积分别是正规矩阵、厄米特矩阵、正定矩阵的条件．推广了Y．E．Kuo的相关结果．另外也给出了两个亚半正定矩阵的张量积还是亚半正定矩阵的充要条件．     

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.     

Generic and typical ranks of multi-way arrays

The concept of tensor rank was introduced in the 20s. In the 70s, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini’s lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries.     

Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, we introduce the notion of Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a new method of simultaneous diagonalizations, we give a complete classification for real hypersurfaces in complex hyperbolic two‐plane Grassmannians with the Reeb parallel Ricci tensor.     

Schroedinger Flows on Compact Hermitian Symmetric Spaces and Related Problems

In this note,we prove that the Schroedinger flow of maps from a closed riemann surface into a compact irreducible Hermitian symmetic space admits a global weak solution.Also,we show the existence of weak solutions to the initial value problem of Heisenberg model with Lie algebra values,which is closely related to the Schroedinger flow on compact Hermitian symmetric spaces.     

Bi-block positive semidefiniteness of bi-block symmetric tensors

The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.     

Symmetric tensor decomposition by alternating gradient descent

The symmetric tensor decomposition problem is a fundamental problem in many fields, which appealing for investigation. In general, greedy algorithm is used for tensor decomposition. That is, we first find the largest singular value and singular vector and subtract the corresponding component from tensor, then repeat the process. In this article, we focus on designing one effective algorithm and giving its convergence analysis. We introduce an exceedingly simple and fast algorithm for rank-one approximation of symmetric tensor decomposition. Throughout variable splitting, we solve symmetric tensor decomposition problem by minimizing a multiconvex optimization problem. We use alternating gradient descent algorithm to solve. Although we focus on symmetric tensors in this article, the method can be extended to nonsymmetric tensors in some cases. Additionally, we also give some theoretical analysis about our alternating gradient descent algorithm. We prove that alternating gradient descent algorithm converges linearly to global minimizer. We also provide numerical results to show the effectiveness of the algorithm.     

Exact solutions of a negative matrix AKNS system with a Hermitian symmetric space

Based on a 4 × 4 matrix Lax pair, we propose a negative matrix AKNS system with a Hermitian symmetric space. A Darboux transformation is constructed by setting a restrictive condition and using the loop group method. The restrictive condition can guarantee the evolution relations of the potential matrices. Using this Darboux transformation and different seed solutions and free parameters, we obtain different types of spatial–temporal distribution structures for various explicit solutions of the negative matrix AKNS system with a Hermitian symmetric space, including the rogue wave, Ma breather, the interaction of two Ma breathers, and parabolic-type soliton solutions.     

Compact Hermitian surfaces of constant antiholomorphic sectional curvatures

Compact Hermitian surfaces of constant antiholomorphic sectional curvatures with respect to the Riemannian curvature tensor and with respect to the Hermitian curvature tensor are considered. It is proved: a compact Hermitian surface of constant antiholomorphic Riemannian sectional curvatures is a self-dual Kaehler surface; a compact Hermitian surface of constant antiholomorphic Hermitian sectional curvatures is either a Kaehler surface of constant (non-zero) holomorphic sectional curvatures or a conformally flat Hermitian surface.

     

A vanishing theorem for modular symbols on locally symmetric spaces

A modular symbol is the fundamental class of a totally geodesic submanifold embedded in a locally Riemannian symmetric space , which is defined by a subsymmetric space . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the -component in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV. Received: December 8, 1996     

Corner view on the crown domain

In this paper we raise a question about the boundary of the crown domain of a Riemannian symmetric space X. In case X is of Hermitian type we give an affirmative answer.     

张量空间中锥的可分解性

在文献 [1 ]的基础上研究张量空间中锥的性质 ,得到了张量空间中射影锥的极端向量的表示形式 .给出了张量空间中射影锥可分解的充分必要条件 ,并由此可得出有限维实空间中真正锥可分解的已有结论 .     

Irreducible function bases of isotropic invariants of a third order three-dimensional symmetric and traceless tensor

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.     

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kähler manifold.     

On Hermitian surfaces with J-invariant Ricci tensor

We prove that a compact Hermitian surface with J-invariant Ricci tensor is K?hler provided that the difference of its scalar and conformal scalar curvature is constant. In particular, there are no locally homogeneous examples of such surfaces with odd first Betti number. Received 20 July 2000.     

Singular values of a real rectangular tensor

Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.     

Necessary and Sufficient Conditions for the Interchange Between Infimum and the Symbol of Integration

We make a study of various notions of decomposability for subsets of measurable functions in relation with the interchange results between infimum and integration. For this we introduce the notions of serial decomposability and of decomposability relatively to an integrand. A characterization of closed serially decomposable subsets of the Lebesgue spaces L _p is given. The second notion of decomposability introduced is characteristic for the interchange property studied. Many examples are presented. The links are made with R. T. Rockafellar’s decomposability, F. Hiai, H. Umegaki’s decomposability, G. Bouchitté and M. Valadier’s stability and normal decomposability introduced by O. Anza Hafsa and J.-P. Mandallena. As applications we obtain exact lower bounds for minimization problems of integral functionals on normally decomposable spaces (spaces of continuous functions for example), and for the minimization of a class of functionals of the Calculus of Variations.     

Biquadratic tensors,biquadratic decompositions,and norms of biquadratic tensors

Biquadratic tensors play a central role in many areas of science.Examples include elastic tensor and Eshelby tensor in solid mechanics,and Riemannian curvature tensor in relativity theory.The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor,respectively.The tensor product operation is closed for biquadratic tensors.All of these motivate us to study biquadratic tensors,biquadratic decomposition,and norms of biquadratic tensors.We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure.Then,either the number of variables is reduced,or the feasible region can be reduced.We show constructively that for a biquadratic tensor,a biquadratic rank-one decomposition always exists,and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition.We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor.Finally,we define invertible biquadratic tensors,and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse,and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor,and the spectral norm of its inverse.     

The Holomorphic 2-Number of a Hermitian Symmetric Space

This paper contains a proof a priori (i.e. independent of the classification of Hermitian symmetric spaces) of a theorem on the holomorphic 2-number of a Hermitian symmetric space. If N=G/K is a Hermitian symmetric space, where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,N) = E₁,... , E_m is the fixed point set of T in N, then for each pair E_i, E_j there is a two-dimensional sphere N_ij N such that E_i and E_j are antipodal points of N_ij.