Math @ Duke

Publications [#304493] of Ezra Miller
Papers Published
 Knutson, A; Miller, E, Gröbner geometry of Schubert polynomials,
Annals of Mathematics, vol. 161 no. 3
(2005),
pp. 12451318, ISSN 0003486X [doi]
(last updated on 2017/12/12)
Abstract: Given a permutation w ∈ S n, we consider a determinantal ideal I w whose generators are certain minors in the generic n × n matrix (filled with independent variables). Using 'multidegrees' as simple algebraic substitutes for torusequivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I w: variously graded multidegrees and Hubert series in terms of ordinary and double Schubert and Grothendieck polynomials; a Gröbner basis consisting of minors in the generic n × n matrix; the StanleyReisner 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 CohenMacaulay, by identifying the StanleyReisner 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 Ktheory 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.


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