Induced operators on symmetry classes of tensors

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.

     

Induced bases of symmetry classes of tensors

Let V^m? denote the mth tensor power of the finite dimensional complex vector space V. Let V_χ(G)?V^m? 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 V^m?. The article considers the corresponding problem of inducing bases of V_χ(G).     

Linear transformation on symmetry classes of tensors

This paper is concerned with linear transformations on a symmetry class of tensors preserving nonzero decomposable elements.     

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 transformation on symmetry classes of tensors

This paper is concerned with linear transformations on a symmetry class of tensors preserving nonzero decomposable elements.     

Quadratic forms on symmetry classes of tensors

Let V:₁,…,V_m be inner product spaces, and let L be a linear transformation on V₁ ?…?V_m 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.     

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.     

Permutation-operator inequalities on certain symmetry classes of tensors

If A∈T(m, N), the real-valued N-linear functions on E_m, and σ∈S_N, the symmetric group on {…,N}, then we define the permutation operator P_σ: T(m, N) → T(m, N) such that P_σ(A)(x₁,x₂,…,x_N = A(x_σ(1),x_σ(2),…, x_σ(N)). Suppose Σ^q_i=1n_i = N, where the n_i are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈P_σ(A₁?A₂???A_q),A₁?A₂?? ? A_q〉 ? 0 for A_i∈S(m, n_i), where S(m, n_i) denotes the subspace of T(m, n_i) consisting of all the fully symmetric members of T(m, n_i). 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 S_N. We let T_G(m, N) denote all A∈T(m, N) such that P_σ(A) = A for all σ∈G. We define the symmetrizer $\text{S}$_G: T(m, N)→T_G(m,N) such that $\text{S}{}_{\mathrm{G}}\text{(A) = 1/|G|Σ}{}_{\mathrm{\sigma \in G}}\text{P}{}_{\mathrm{\sigma }}\text{(A)}$. Suppose H is a subgroup of G and A∈T_H(m, N). Clearly $\text{6}\text{S}{}_{\mathrm{<mtext>G</mtext>}}\text{6(A) 6? 6A6.}$ We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of T_H(m,N), then can we explicitly present a constant C>0 such that $\text{6}\text{S}{}_{\mathrm{G}}\text{(A)6?C6A6}$ for all A∈D?     

Dimensionality of algebras of symmetry classes of tensors

In this paper we give necessary and sufficient conditions for Algebras of Symmetry Classes of Tensors to be finite dimensional.     

Pattern inventories associated with symmetry classes of tensors

An extension of Polya's Theorem inventories the sequence set corresponding to an induced basis in a higher degree symmetry class of tensors.     

Elementary divisors of induced transformations on symmetry classes of tensors

Let V_χ(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*...*v_m 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 V_χ(G), i.e., K(T)v₁*v₂*...*v_m=Tv₁*Tv₂*...*Tv_m. Denote by D(T) the derivation operator D(T)v₁*v₂*...*v_m=Tv₁*v₂...*v_m+v₁*Tv₂*v₃* ...*v_m+...+v₁*v₂*...*v_m–1*Tv_m. The article concerns the elementary divisors of K(T) and D(T).     

The subspaces and orthonormal bases of symmetry classes of tensors

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.     

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.     

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.     

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.     

Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group

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.     

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.