Nonzero decomposable symmetrized tensors

Suppose k₁ ? ? ? k_t ? 1, m₁ ? ?? m_r ? 1, k₁+ ? +k_t = m₁+ ? +m_r = m. Let λ=(k₁,…,k_t) be a character of the symmetric group S_m. The restriction of λ to S_m1X…XS_mr contains the principal character as a component if and only if λ majorizes (m₁,…,m_r). This result is used to characterize the index set of the nonzero decomposable symmetrized tensors, corresponding to S_m and λ, which are induced from a basis of the underlying vector space.     

Orthogonal decomposable symmetrized tensors

In this article we present a sufficient condition for orthogonality of decompassable symmetrized tensors.     

Linear preservers of decomposable symmetric tensors

Let U be an n-dimensional vector space over an algebraically closed field F. Let U(^m) denote the mth symmetric power of U. For each positive integer k≤min{m,n}, let D_k denote the set of all nonzero decomposable elements x₁ …x_m in U(^m) such that dim(x₁ …x_m ) = k and E_k denote the set of all decomposable elements x₁ …x_m in U(^m) such that dim(x₁ …x_m ) ≤ k. In this paper we first show that E_k is an algebraic variety with D_k as a dense subset and determine the dimension of E_k . We next use these results to study the structure of linear mappings T on U^m such that T(D_k ) ? D_k or T(E_k ) ? E_k for some fixed k.     

Additive preservers of non-zero decomposable tensors

Let T be an additive mapping from a tensor product of vector spaces over a field into itself. We describe T for the following two cases: (i) T is surjective and sends non-zero decomposable elements to non-zero decomposable elements, and (ii) T(A) is a non-zero decomposable element if and only if A is a non-zero decomposable element.     

Conditions for equality of decomposable symmetric tensors

Let x₁,…,x_m be linearly independent vectors. We give a necessary and sufficient condition for $\text{x}{}_1\text{1…1x}{}_{\mathrm{m}}\text{=y}{}_1\text{1…1y}{}_{\mathrm{m}}$ to hold. Some consequences of this result are also presented.     

A high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors

We present a high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors, and this is a counter-example of the claim presented in [1].     

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.     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

Conditions for equality of decomposable symmetric tensors. II.

We derive consequences of a condition for the equality of two star products given by the second author. We also study another method for the same problem which consists of comparing the components, in an appropriate basis, of the star products involved.     

Some necessary and sufficient conditions satisfied by decomposable tensors in grassmann spaces

The matrix form of the quadratic Plücker relations for decomposable tensors in Grassmann spaces is presented and some equivalent conditions are obtained. The proofs of the equivalence are elementary. The same methods are used to prove an analogous result for subspace incidence.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

Twistors and Cartan Connections

We give a simple, general approach for constructing a twistorspace Z with a CR structure for a manifold Mwith a geometric structure. The starting point is a principalG-bundle : P M with a Cartanconnection. The twistor space is an appropriate subgroup ofG. The integrability condition for the CR structure isdescribed in terms of the curvature of the Cartan connection.     

Cartan Geometries and Dynamics

We show how some basic dynamical ideas can be brought to bear on the study of Cartan geometries. In our main results, we give conditions for certain types of Cartan geometries to have constant curvature. We also consider ergodic actions of `higher-rank' semisimple groups on bundles supporting (not necessarily invariant) Cartan connections and show that these are `standard locally homogeneous' actions provided that some noncompact 1-parameter subgroup `preserves' a Cartan geometry.     

Cartan decompositions and BGG-resolutions

Necessary and sufficient criteria are given for the existence of BGG-resolutions (finite resolutions of modules by finite direct sums of Weyl modules) for simple modules over quasi-hereditary algebras, which have strong exact Borel subalgebras and strong Δ-subalgebras. Our main technical tool is the existence of Cartan decompositions for these algebras. The results apply to simple objects in the BGG-categoryO of a finite-dimensional semisimple complex Lie algebra and to finite dimensional simple rational modules over simply connected semisimple algebraic groups.