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1.
Chi-Kwong Li Alexandru Zaharia 《Transactions of the American Mathematical Society》2002,354(2):807-836
Let be an -dimensional Hilbert space. Suppose is a subgroup of the symmetric group of degree , and is a character of degree 1 on . Consider the symmetrizer on the tensor space
defined by and . The vector space
is a subspace of , called the symmetry class of tensors over associated with and . The elements in of the form are called decomposable tensors and are denoted by . For any linear operator acting on , there is a (unique) induced operator acting on satisfying
In this paper, several basic problems on induced operators are studied.
defined by and . The vector space
is a subspace of , called the symmetry class of tensors over associated with and . The elements in of the form are called decomposable tensors and are denoted by . For any linear operator acting on , there is a (unique) induced operator acting on satisfying
In this paper, several basic problems on induced operators are studied.
2.
Let Vm? denote the mth tensor power of the finite dimensional complex vector space V. Let Vχ(G)?Vm? be the symmetry class of tensors corresponding to the permutation group G and the irreducible character χ of G. Each basis of V induces, in a natural way, a basis of Vm?. The article considers the corresponding problem of inducing bases of Vχ(G). 相似文献
3.
M. H. Lim 《Linear and Multilinear Algebra》1976,3(4):267-280
This paper is concerned with linear transformations on a symmetry class of tensors preserving nonzero decomposable elements. 相似文献
4.
M. H. Lim 《Linear and Multilinear Algebra》1982,12(2):109-123
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. 相似文献
5.
M.H. Lim 《Linear and Multilinear Algebra》2013,61(4):267-280
This paper is concerned with linear transformations on a symmetry class of tensors preserving nonzero decomposable elements. 相似文献
6.
Herbert A. Robinson 《Linear and Multilinear Algebra》2013,61(4):233-241
Let V:1,…,Vm be inner product spaces, and let L be a linear transformation on V1 ?…?Vm which satisfies (Lz,z)=0 for every decomposable tensor z. It is known that if the field is the complex numbers, then (Lz,z)=0 for every z. This paper contains a short proof of this result, an extension of it to arbitrary symmetry classes of tensors, and an analysis of its failure when the field is the real numbers. 相似文献
7.
M. H. Lim 《Linear and Multilinear Algebra》2013,61(2):109-123
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. 相似文献
8.
If A∈T(m, N), the real-valued N-linear functions on Em, and σ∈SN, the symmetric group on {…,N}, then we define the permutation operator Pσ: T(m, N) → T(m, N) such that Pσ(A)(x1,x2,…,xN = A(xσ(1),xσ(2),…, xσ(N)). Suppose Σqi=1ni = N, where the ni are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈Pσ(A1?A2???Aq),A1?A2?? ? Aq〉 ? 0 for Ai∈S(m, ni), where S(m, ni) denotes the subspace of T(m, ni) consisting of all the fully symmetric members of T(m, ni). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of SN. We let TG(m, N) denote all A∈T(m, N) such that Pσ(A) = A for all σ∈G. We define the symmetrizer G: T(m, N)→TG(m,N) such that . Suppose H is a subgroup of G and A∈TH(m, N). Clearly We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of TH(m,N), then can we explicitly present a constant C>0 such that for all A∈D? 相似文献
9.
10.
J. Hillel 《Linear and Multilinear Algebra》1976,4(1):41-44
In this paper we give necessary and sufficient conditions for Algebras of Symmetry Classes of Tensors to be finite dimensional. 相似文献
11.
An extension of Polya's Theorem inventories the sequence set corresponding to an induced basis in a higher degree symmetry class of tensors. 相似文献
12.
Let Vχ(G) denote the symmetry class of tensors over the vector space V associated with the permutation group G and irreducible character χ. Write v1*v2*...*vm for the decomposable symmetrized product of the indicated vectors (m=degG). If T is a linear operator on V, let K(T) denote the associated operator on Vχ(G), i.e., K(T)v1*v2*...*vm=Tv1*Tv2*...*Tvm. Denote by D(T) the derivation operator D(T)v1*v2*...*vm=Tv1*v2...*vm+v1*Tv2*v3* ...*vm+...+v1*v2*...*vm–1*Tvm. The article concerns the elementary divisors of K(T) and D(T). 相似文献
13.
14.
We present here the dimensions of some subspaces in the symmetry classes of tensors and some methods for constructing orthonormal bases of the symmetry classes of tensors. 相似文献
15.
Russell Merris 《Linear and Multilinear Algebra》2013,61(3):171-178
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. 相似文献
16.
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. 相似文献
17.
Russell Merris 《Linear and Multilinear Algebra》1978,6(3):171-178
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. 相似文献
18.
19.
In this article necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the dihedral group. 相似文献
20.
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. 相似文献