Bases for Spaces of Highest Weight Vectors in Arbitrary Characteristic

Let k be an algebraically closed field of arbitrary characteristic. First we give explicit bases for the highest weight vectors for the action of GLr ×GLs on the coordinate ring k[Mat m rs ] of m-tuples of r × s-matrices. It turns out that this is done most conveniently by giving an explicit good GLr ×GLs-filtration on k[Mat m rs ]. Then we deduce from this result explicit spanning sets of the k[Mat n ] GL n -modules of highest weight vectors in the coordinate ring k[Matn] under the conjugation action of GLn.