Modular symmetry classes of tensors

We introduce the notion of modular symmetry classes of tensors and give a necessary and sufficient condition for a modular symmetry class of tensors associated with the full symmetric group to be non-zero. Then we use modular symmetry classes of tensors to study the polynomial representations of GL(V), where V is a vector space over a field of characterisitic p. At the end we introduce a non-degenerate bilinear form on a modular symmetry class. Some problems are also given.