A determinantal formula for the exterior powers of the polynomial ring Dan LaksovAnders Thorup 14N1514M1505E05determinant formulaSchubert calculusGatto's formulaexterior algebrasGiambelli's formulaGrassman schemessymmetric functionssymetrizing operatorsdivided difference operatorsintersection theoryuniversal splitting algebras We present a very general, conceptually natural, explicit and computationally efficient \emph{Schubert calculus}. It consists of two strongly interrelated parts, a structure theorem for an exterior power of the polynomial ring in one variable as a module over the ring of symmetric polynomials and a determinantal formula. Both parts are similar to corresponding results in algebra, combinatorics and geometry. To emphasize the connections with these fields we give several proofs of our main results, each proof illuminating the theory from a different algebraic, combinatorial, or geometric angle. The main application of our theory is to the Schubert calculus of Grassmann schemes, where it gives a natural homology and cohomology theory for the Grassmannians in a very general setting. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.2937 10.1512/iumj.2007.56.2937 en Indiana Univ. Math. J. 56 (2007) 825 - 846 state-of-the-art mathematics http://iumj.org/access/