Gröbner Geometry of Schubert Polynomials 1247

Given a permutation w ∈ Sn, we consider a determinantal ideal Iw whose generators are certain minors in the generic n × 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 Iw: • variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials; • a Gröbner basis consisting of minors in the generic n× n matrix; • the Stanley–Reisner simplicial complex of the initial ideal in terms of known combinatorial diagrams [FK96], [BB93] associated to permutations in Sn; and • a procedure inductive on weak Bruhat order for listing the facets of this complex. We show that the initial ideal is Cohen–Macaulay, by identifying the Stanley– Reisner complex as a special kind of “subword complex in Sn”, 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. Our main theorems provide a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. More precisely, we apply these theorems to: • define 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; *AK was partly supported by the Clay Mathematics Institute, Sloan Foundation, and NSF. EM was supported by the Sloan Foundation and NSF. 1246 ALLEN KNUTSON AND EZRA MILLER • rederive from a topological perspective Fulton’s Schubert polynomial formula for universal cohomology classes of degeneracy loci of maps between flagged vector bundles; • supply new proofs that Schubert and Grothendieck polynomials represent cohomology and K-theory classes on the flag manifold; and • provide determinantal formulae for the multidegrees of ladder determinantal rings. The proofs of the main theorems introduce the technique of “Bruhat induction”, consisting of a collection of geometric, algebraic, and combinatorial tools, based on divided and isobaric divided differences, that allow one to prove statements about determinantal ideals by induction on weak Bruhat order.