Gröbner geometry of Schubert polynomials through ice

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson Miller (2005) via antidiagonal Gr\"obner degeneration matrix varieties. We consider instead diagonal degenerations. In this dual setting, Knutson, Miller, Yong (2009) obtained alternative combinatorics for the class "vexillary'' initiate a study general degenerations, relating them to neglected formula Lascoux (2002) in terms $6$-vertex ice model (recently rediscovered Lam, Lee, Shimozono (2018) guise "bumpless dreams'').