Structure Constants for Bases of the Polynomial Ring

University Of Southern California, Los Angeles CA

Investigators

Abstract

Schubert calculus began around 1879 with Herman Schubert asking, and in special cases answering, enumerative questions in geometry. For example, how many points in the plane meet two given lines simultaneously or how many lines in space meet four given lines? To answer the latter, Schubert considered the case where the first line intersects the second and the third intersects the fourth, in which case the answer is 2 (the line connecting the two points of intersection and the line of intersection of the two planes spanned by the two pairs of intersecting lines). He then asserted, by his principle of conservation of number, that the general answer, if finite, must also be 2. David Hilbert, in his 15th problem for the twentieth century, set out the task of making rigorous Schubert's principle of conservation of number. Cohomology resolved this and lead us to modern Schubert calculus and intersection theory, which has ramifications in geometry, topology, combinatorics, and even plays a central role in string theory. Yet today, Schubert's original question remains unanswered: how can one effectively compute intersection numbers for linear subspaces? Lascoux and Schutzenberger in 1985 defined polynomial representatives for the Schubert classes in the cohomology ring of the complete flag variety. These Schubert polynomials give explicit polynomial representations of the Schubert classes whose structure constants enumerate flags (sequences of nested linear subspaces) in a suitable triple intersection of Schubert varieties, which are precisely the nonnegative integers Schubert calculus aims to compute. The principal investigator will lift powerful tools and techniques from symmetric function theory to the full polynomial ring and apply them to Schubert polynomials with the ultimate goal of finding a combinatorial formula for these intersection numbers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.