Mathematics Colloquium

Schubert calculus began in the 1880s when Hermann Schubert began asking enumerative questions in geometry, such as how many lines in space are incident to four given lines. Efforts to build a rigorous foundation for these questions led to the development of cohomology rings and modern intersection theory. In the 1980s, Lascoux and Schutzenberger defined an explicit basis for polynomials, called Schubert polynomials, whose structure constants precisely compute these intersection numbers. For the special case of the grassmannian sub variety of the complete flag manifold, these polynomials are Schur polynomials and the classical Littlewood—Richardson rule gives their structure constants by enumerating Yamanouchi tableaux, which are certain ways of putting numbers into boxes. In this talk, I'll survey combinatorial models for Schubert polynomials that generalize these tableaux models for Schur polynomials and lead to new cases for computing structure constants by enumerating new ways of putting numbers in boxes.