Gröbner geometry of Schubert polynomials


Given a permutation $w \in S_n$, we consider a determinantal ideal $I_w$ whose generators are certain minors in the generic $n \times n$ 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 Gröbner basis consisting of minors in the generic $n \times n$ matrix;
  • the Stanley–Reisner simplicial complex of the initial ideal in terms of known combinatorial diagrams [FK96], [BB93] associated to permutations in $S_n$; 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 $S_n$”, 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.

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;
  • 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.


Allen Knutson

Department of Mathematics, University of California, Berkeley, CA 94720, United States

Ezra Miller

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States