Gröbner geometry of Schubert polynomials

Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torus-equivariant cohomology classes of certain determinantal loci in the vector space of n×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torus-equivariant cohomology class of a component (a scheme-theoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant K-theory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in K-theory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are the facets of a simplicial complex, are in an obvious bijection with the rcgraphs of Fomin and Kirillov [FK96b]. Thus our positive formula for Schubert polynomials agrees with (and provides a geometric proof of) the combinatorial formula of Billey-Jockusch-Stanley [BJS93]. Moreover, we shell this complex (as one of a new class of vertex-decomposable complexes we introduce), which shows that the initial ideal of the minors is a Cohen–Macaulay Stanley–Reisner ideal. This provides a new proof that Schubert varieties are Cohen–Macaulay. The multidegree of any finitely generated multigraded module, defined here based on torus-equivariant cohomology classes, generalizes the usual Z-graded degree to finer gradings. Part of the Gröbner basis theorem includes formulae for the multidegrees and Hilbert series of determinantal ideals in terms of Schubert and Grothendieck polynomials. In the special case of vexillary determinantal loci, which include all one-sided ladder determinantal varieties, the multidegree formulae are themselves determinantal, and our new antidiagonal Gröbner basis statement contrasts with known diagonal Gröbner basis statements. Interpreting the Schubert polynomials as equivariant cohomology classes on matrices gives a topological reason (see also [FR01]) why Schubert polynomials are the characteristic classes for degeneracy loci [Ful92]: the mixing space AK was partly supported by the Clay Mathematics Institute, Sloan Foundation, and NSF. EM was supported by the Sloan Foundation and NSF.