置换群直积的张量对称类的秩

假定基环R是特征为零的整环，并且使得它上每个有限生成的投射模是自由模．本文研究有限秩自由R-模的张量积相对于有限个置换群直积而言的张量对称类，给出了张量对称类非平凡的判别准则以及相应张量对称类秩之间的关系式，并将所得结果应用到模情形．     

Symmetry classes of tensors associated with certain groups

We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with certain subgroups of the full symmetric group The dimensions of these symmetry classes of tensors are also given     

Symmetry classes of tensors and semi-direct product of finite abelian groups

In the study of symmetry classes of tensors, finding examples of symmetry classes of tensors that possess an o*-basis is of considerable interest. There are only few classes of groups that have been provided a necessary and sufficient condition for having such a basis. There is no general criterion for any finite group yet. In this note, we provide a necessary and sufficient condition for the existence of o*-basis of symmetry classes of tensors associated with semi-direct product of some finite abelian groups and, consequently, their wreath product.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

Construction of orthonormal bases in higher symmetry classes of tensors

We present a method for constructing an orthonormal basis for a symmetry class of tensors from an orthonormal basis of the underlying vector space. The basis so obtained is not composed of decomposable symmetrized tensors. Indeed, we show that, for symmetry classes of tensors whose associated character has degree higher than one, it is impossible to construct an orthogonal basis of decomposable symmetrized tensors from any basis of the underlying vector space. We end with an open problem on the possibility of a symmetry class having an orthonormal basis of decomposable symmetrized tensors.     

Tabulation of the number of independent components of physical tensors for the magnetic classes

Following the method of Bhagavantam and Pantulu⁸ the number of independent components of tensors up to 4th rank is derived for the magnetic classes. This method is extended by Lyubarskii¹ to cases of tensors having intrinsic symmetry. The results cover all cases of tensors lacking intrinsic symmetry as well as of tensors having the symmetries of Table II, and are in agreement with previous results.^{5, 8}     

The structure of higher degree symmetry classes of tensors II

Various results bearing on the construction of bases for symmetry classes of tensors are discussed. Applications are given to the production of explicit homogeneous polynomial representations of the full linear group.     

The structure of higher degree symmetry classes of tensors II

Various results bearing on the construction of bases for symmetry classes of tensors are discussed. Applications are given to the production of explicit homogeneous polynomial representations of the full linear group.     

Applications of spherical functions in multilinear algebra

Spherical functions are shown to organize and unify previous work on generalized matrix functions and symmetry classes of tensors. At the same time, new areas of investigation are suggested.     

Linear mappings on symmetry classes of tensors

In this paper we give a survey of results concerning linear mappings on symmetry classes of tensors that preserve decomposable elements and its related topic about linear mappings on spaces of matrices that preserve a fixed rank.     

Linear mappings on symmetry classes of tensors

In this paper we give a survey of results concerning linear mappings on symmetry classes of tensors that preserve decomposable elements and its related topic about linear mappings on spaces of matrices that preserve a fixed rank.     

On geometry of weakly cosymplectic manifolds

We consider classes of weakly cosymplectic manifolds whose Riemannian curvature tensors satisfy contact analogs of the Riemannian–Christoffel identities. Additional properties of the Riemannian curvature tensor symmetry are found and a classification of weakly cosymplectic manifolds is obtained.     

-inclusion sets for eigenvalues of a tensor

In this paper, we mainly focus on new inclusion sets for eigenvalues of a tensor. First, we propose new inclusion sets for eigenvalues of a tensor, which are sharper than some existing inclusion sets, and obtain the law of distribution of the number of eigenvalues for a tensor. Second, two new classes of tensors are introduced. Third, some bounds on the spectral radii for nonnegative tensors are given. Fourth, some checkable sufficient conditions for the positive definiteness (positive semidefiniteness) of some classes of even-order real symmetric tensors are obtained.     

Lower bounds for the norms of decomposable symmetrized tensors

Lower bounds are given for the difference of two decomposable symmetrized tensors. The first bound uses a norm which makes the component vectors in a decomposable symmetrized tensor part of an orthonormal basis. The second bound holds only for decomposable elements of symmetry classes whose associated characters are linear.     

(k)-Characters and the triviality of symmetry classes

The author introduces the notion of (k)-character and its application to symmetry classes of tensors. A necessary and sufficient condition for the symmetry class V_λ(G) to be trivial in the case degree of G ? 2dim V has also been found.     

A note on the Stokes resistance tensors of multiparticle systems in shear fields

The Stokes resistance tensors (material tensors) of a multiparticle system subjected to a general shear flow depends on the spatial configuration of the particles, their shape, size and location of some arbitrarily fixed set of points locked to the particles by which their linear velocities are traced. The objective of this work is the determination of some symmetry properties of the material tensors and the transformation laws which govern them under translation of the set of points fixed to the particles.     

Irreducible representations of the full linear group

We obtain some results on Young diagrams. Based on these results, we construct bases for Young symmetry classes of tensors. Using these bases, we obtain a complete reduction of the representation A ??^mA [A∈GL(n,$\text{C}$] and irreducible matrix representations of the full linear group.     

Higher-order elasticity and strength surface tensors in the two-dimensional case

The number of independent components of the higher-order elasticity and strength surface tensors has been calculated for the two-dimensional case. Media with different symmetry classes are considered.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 372–374, March–April, 1974.