Schubert calculus

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In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest; the phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties. Even more generally, “Schubert calculus” is often understood to encompass the study of analogous questions in generalized cohomology theories.

The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety.

The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring.

In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (by block matrix).

Putting Schubert's system on a rigorous footing is Hilbert's fifteenth problem.


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