tailieunhanh - Đề tài "Gr¨obner geometry of Schubert polynomials "

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; | Annals of Mathematics Gr obner geometry of Schubert polynomials By Allen Knutson and Ezra Miller Annals of Mathematics 161 2005 1245 1318 Grobner geometry of Schubert polynomials By Allen Knutson and Ezra Miller Abstract Given a permutation w E Sn we consider a determinantal ideal Iw 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 Iw variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials a Grobner basis consisting of minors in the generic n X 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 .