张量积空间线性秩一保持映射

Linear rank one preserving maps on tensor product space

令F是特征为0的代数闭域,V1,…,Vm是n维F-向量空间。R1是张量积空间■中所有秩一张量全体。本文主要给出以下结果:线性映射T:■满足T(R1)?R1的充分必要条件是存在可逆线性变换Ai∈Vi,i∈[1,m],使得T(X)=(Aπ(1),Aπ(2),…,Aπ(m))·Xπ,?π∈Sm对于任意■成立,其中■,Sm是m元对称置换群。

Let F be an algebraically closed field of characteristic 0 and V1,…,Vm be F-vector spaces with dimension n.Let R1 denote the set of all rank one tensors in the tensor product space■.In this paper,it is shown that a linear map T:■satisfies T(R1)?R1if and only if there exist nonsingular linear transformations Ai on Vi,i∈[1,m]such that T(X)=(Aπ(1),Aπ(2),…,Aπ(m))·Xπ,?π∈Smfor all■,where■and Sm is a symmetric group of permutations on m elements.