Gröbner geometry of Schubert polynomials through ice (Q5918494)

The so-called Schubert polynomials are significant in combinatorics and in geometry. There are two combinatorial ways of writing Schubert polynomials as sums of products of linear factors. One is parameterized by ``pipe dreams'' [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)], the other is parameterized by ``bumpless pipe dreams'' [\textit{A. Lascoux}, ``Chern and Yang through ice'', Preprint; \textit{T. Lam} et al., Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)]. In geometry Schubert polynomials are fundamental classes of matrix Schubert varieties [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008); Knutson-Miller, loc. cit.]. Under favorable (e.g. Gröbner) degenerations the fundamental class of a variety does not change. Hence if a matrix Schubert variety is degenerated to a union of linear spaces, then its fundamental class is written as a sum of product of linear factors. Such geometric interpretation for the ``pipe dream'' formula is known, via the anti-diagonal Gröbner degeneration. The paper under review searches for such a geometric interpretation of the ``bumpless pipe dream'' formula for Schubert polynomials. Namely, the authors conjecture that the ``bumpless pipe dream'' formula corresponds naturally to any diagonal Gröbner degeneration. They prove this conjecture for a large class of permutations they call banner permutations. Basic steps of their recursive arguments rely on Lascoux-Schützenberger transitions.