Bases for Fixed Points of Unipotent Elements Acting on the Tensor Square and the Spaces of Alternating and Symmetric 2-Tensors

If V is a vector space over a field K, then an element g of the general linear group GL(V) acts on V V, on the space of alternating 2-tensors A(V), and on the space of symmetric 2-tensors S(V). For a unipotent element g, we exhibit bases for the subspace of fixed points of g acting on both V V and A(V) which are valid for every field K, and a basis for the subspace of fixed points of g acting on S(V) which is valid for every field K with char K ≠ 2.