Gr\"obner geometry of Schubert polynomials

TL;DR

This paper provides a geometric framework for understanding Schubert polynomials, establishing their positivity, and describing their algebraic and combinatorial properties through determinantal ideals, multidegrees, and shellable complexes.

Contribution

It introduces a unified geometric approach to Schubert polynomials, including Gr"obner bases, multidegrees, and combinatorial complexes, with proofs of positivity and Cohen-Macaulay properties.

Findings

Describes multidegrees and Hilbert series in terms of Schubert polynomials.

Establishes a Gr"obner basis of minors for determinantal ideals.

Proves the initial ideal is Cohen-Macaulay and shellable.

Abstract

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. Given a permutation w in S_n, we consider a determinantal ideal I_w whose generators are certain minors in the generic n x n matrix (filled with independent variables). Using `multidegrees' as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I_w: - variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials; - a Gr\"obner basis consisting of minors in the generic n x n matrix; - the Stanley-Reisner complex of the initial ideal in terms of known combinatorial diagrams associated to permutations in S_n; and - a procedure inductive on weak Bruhat order for listing the facets of this complex, thereby generating the coefficients of Schubert polynomials by a positive recursion on combinatorial diagrams. We show that the initial ideal is Cohen-Macaulay, by identifying the Stanley-Reisner complex as a special kind of ``subword complex in S_n'', which we define generally for arbitrary Coxeter groups, and prove to be shellable by giving an explicit vertex decomposition. We also prove geometrically a general positivity statement for multidegrees of subschemes.