Schubert calculus has its origins in enumerative questions asked by the geometers of the 19th century, such as “how many lines meet four fixed lines in three-space?”  These problems can be recast as questions about the structure of cohomology rings of geometric spaces such as flag varieties.  Borel’s isomorphism identifies the cohomology of the complete flag variety with a simple quotient of a polynomial ring.  Lascoux and Schützenberger (1982) defined Schubert polynomials, which are coset representatives for the Schubert basis of this ring.  However, it was not clear if this choice was geometrically natural.  Knutson and Miller (2005) provided a justification for the naturality of Schubert polynomials via antidiagonal Gröbner degenerations of matrix Schubert varieties. Furthermore, they showed that pre-existing combinatorial objects called pipe dreams govern this degeneration.

In this talk, we consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics in the special case of vexillary matrix Schubert varieties. We will discuss general diagonal degenerations, relating them to older work of Lascoux (2002) on the 6-vertex ice model. The cohomological version of Lascoux’s formula was recently rediscovered by Lam, Lee, and Shimozono (2018) as “bumpless pipe dreams.” We will explain this connection and discuss progress towards understanding diagonal Gröbner degenerations of matrix Schubert varieties in the general setting.

Website: http://www-personal.umich.edu/~weigandt/