Gröbner geometry of Schubert polynomials

By Allen Knutson, Ezra Miller. 2001

393 citations recorded by the source corpus.

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 ?emultidegrees?f as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal Iw.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/41dbd70ed76e29ca8b4f332f84865e69c69be279

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Citations in the corpus

GOVERNING SINGULARITIES OF SCHUBERT VARIETIES (2006, background): “After re-writing our matrices upside down, this conjecture would generalize the conclusion for matrix Schubert varieties found in [20].”
Gröbner geometry of vertex decompositions and of flagged tableaux (2005, methodology): “The “only if” direction is striking because the essential minors in Iπ form a Gröbner basis for any antidiagonal term order, even if the permutation π is not vexillary [KM1].”
Logarithmic concavity of Schur and related polynomials (2019, background): “8.2], the fundamental class of Yin the Chow group of Xis given by rYsS wpH 1;:::;H nqXrXs; H ic 1pˇ i Op1qq: An alternative proof of the displayed formula, in a more reﬁned setting, was obtained in [KM05] through an explicit degeneration of Y. An important point for us is that Y is irreducible of ex”
Back stable Schubert calculus (2018, background): “n×2n). But Y′ is a product. The equivariant class of the aﬃne space B− is the product of the weights of the matrix entries that are set to zero in B− and the equivariant class of Mw is γ−xnSw(x;a) by [KM] (the shift in xvariables is due to the convention on weights). We deduce that [Y′] = Y 1−n≤i&”
Schubert polynomials and classes of Hessenberg varieties (2007, methodology): “with this monomial form correspond to dominant permutations. To see this, we use the combinatorics of pipe dreams, which give a combinatorial model for the terms in a Schubert polynomial. See [MS] or [KM] for details and examples. More precisely, a pipe dream in Sn is a subset of boxes from the n×n ”
Grothendieck polynomials via permutation patterns and chains in the Bruhat order (2004, neutral): “by the NSF VIGRE program. Research of Sottile supported in part by NSF CAREER grant DMS-0134860, the Clay Mathematical Institute, and the MSRI. 1 2 CRISTIANLENART,SHAWNROBINSON,ANDFRANKSOTTILE Miller [19, 20] gave a geometric and commutative algebraic construction of Grothendieck polynomials, recove”
Four positive formulae for type A quiver polynomials (2003, neutral): “Writing double Schubert polynomials in terms of pipe dreams [FK96] then provides another geometric formula for double quiver polynomials, via [ KM03a ].”
A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (2007, background): “There has been significant interest in the Grothendieck ring of X and of related varieties, see work on, e.g., quiver loci [Bu02b, Bu05b, Mi05, BuKrShTaYo06], Hilbert series of determinantal ideals [ KnMi05 , KnMiYo05a, KnMiYo05b], applications to invariants of matroids [Sp06], and in relation to re”
Excited Young diagrams and equivariant Schubert calculus (2007, methodology): “(2) Combinatorics used in [7], [3], [ 13 ], [14] are closely connected with EYDs.”
Double Schubert polynomials for the classical groups (2008, methodology): “More recently, two related constructions connecting double Schubert polynomials to equivariant cohomology of flag manifolds, using either Thom polynomials or Gröbner degenerations were obtained independently by Fehér and Rimányi [10] and by Knutson and Miller [26].”
Brick polytopes of spherical subword complexes and generalized associahedra (2011, background): “ubword complexes of type A(see also a generalization developed in [SS12]). Such subword complexes were introduced by A. Knutson and E. Miller in the context of Grobner geometry of Schubert varieties [KM05]. The denition was then extended to all Coxeter groups in [KM04]. They proved that these simpli”
Subword complexes, cluster complexes, and generalized multi-associahedra (2011, background): “Miller, first in type A to study the combinatorics of determinantal ideals and Schubert polynomials [35], and then for all Coxeter groups in [34].”
Toric degeneration of Schubert varieties and Gelfand¿Tsetlin polytopes (2003, methodology): “Therefore—and this is the main point—applying the combinatorial characterization of degenerated matrix Schubert varieties in [ KM03 ], we characterize in Theorem 8 which faces of the Gel ′ fand–Cetlin toric variety occur”