Derivatives compute integrals

Let B := Q[x1, x2, . . .] be a polynomial ring in infinitely many indeterminates over the rationals, where each indeterminate xi is given weight i. Computing partial derivatives of polynomials in B amounts to play Schubert Calculus on the Sato’s Universal Grassmann Manifold (the inductive limit of finite dimensional Grassmannians with respect to obvious inclusions maps). In particular the d-th iterated of the partial derivative with respect to x1 of a weighted homogeneous polynomial of degree d amounts to compute an integral (i.e. the degree of a top codimension intersection cycle) on a finite dimensional grassmannian (that explains the title of the talk). We will illustrate how this idea works to obtain an efficient compact formula for the Plücker degree of a Schubert variety, deducing the generating function for them (existing in the classical theory of symmetric functions to count Young tableaux of a given shape) and, time permitting, showing a generating function encoding all the possible integrals of products of special Schubert cycles on finite dimensional grassmannians.