Complements of tableaux and straightening bideterminants

In this paper, we study the combinatorics related to complements of tableaux. Using the representation theory of general linear groups and a known construction of the dual Weyl modules, we obtain a new straightening algorithm for bideterminants and prove that it is compatible with the complement construction of tableaux. As an application, we get a purely combinatorial construction of the isomorphisms between generalized (classical) Schur algebras introduced first by Henke, Koenig and the author.