Gröbner geometry of Schubert polynomials.

*(English)*Zbl 1089.14007The manifold \(\mathcal Fl_n\) of complete flags in the \(n\) dimensional vector space \(\mathbb C^n\) over the complex numbers is an object that, by its various definitions, is an object in the intersection of algebra and geometry. On the one hand it can be expressed as the quotient \(B\backslash \text{GL}_n\) of all invertible \(n\times n\)-matrices by its subgroup of lower triangular matrices, and on the other hand as fibers of certain bundles constructed universally from complex vector bundles. Combinatorics enter into the study via the cohomology ring \(H^\ast(\mathcal Fl_n) =H^\ast(\mathcal Fl_n;\mathbb Z)\) with integer coefficients, that can be described as the quotient of the polynomial ring \(\mathbb Z[x_1,\dots, x_n]\) modulo the ideal generated by all non-constant homogeneous functions invariant under permutations of \(x_1,\dots, x_n\). This ring is a free abelian group of rank \(n!\) with basis given by monomials dividing \(\prod_{i=1}^{n-1}x_i^{n-i}\). The ring also has a much more geometric basis given by the Schubert classes \([X_w]\) in the cohomology ring \(H^\ast(\mathcal Fl_n)\).

This article makes an important contribution to bridging the algebra and combinatorics of Schubert polynomials with the geometry of Schubert varieties. It brings new perspectives to problems in commutative algebra concerning ideals generated by minors of generic matrices, and provides a geometric context in which polynomial representatives for Schubert classes are uniquely singled out with no choices but a Borel subgroup of the general linear group \(\text{GL}_n\mathbb C\) in such a way that it is geometrically obvious that these representatives have nonnegative coefficients.

One of the main ideas in the article is to translate ordinary cohomological statements concerning Borel orbit closures on the flag manifold \(\mathcal Fl_n\) into equivariant-cohomological statements concerning double Borel orbit closures on the \(n\times n\) matrices \(M_n\). To be more precise, the preimage \(\widetilde X_w\subseteq \text{GL}_n\mathbb C\) of a Schubert variety \(X_w\in \mathcal Fl_n\) is an orbit closure for the action of the product \(B\times B^+\) of the lower and upper triangular subgroups of \(\text{GL}_n\mathbb C\) acting by multiplication on the left and right. When \(\overline X_w\subseteq M_n\) is the closure of \(\widetilde X_w\) and \(T\) is the torus in \(B\), the \(T\)-equivariant cohomology class \([\overline X_w]_T\in H_T^\ast(M_n)\) is the polynomial representative. It has positive coefficients because there is a \(T\)-equivariant flat (Gröbner) degeneration of \(\overline X_w\) to \(\mathcal L_w\) that is a union of coordinate subspaces \(L\subseteq M_n\). Each subspace \(L\subseteq \mathcal L_w\) has an equivariant cohomology class \([L]_T\in H_T^\ast(M_n)\) that is a monomial in \(x_1,\dots, x_n\), and the sum of these is \([\overline X_w]_T\). The formula is \([\overline X_w]_T = [\mathcal L_w]_T =\sum_{L\in \mathcal L_w}[L]_T\). More importantly, the authors identify a particularly natural degeneration of the matrix Schubert variety \(\overline X_w\) with a reduced and Cohen-Macaulay limit \(\mathcal L_w\) in which the subspaces have combinatorial interpretations and coincides with known combinatorial formulas for Schubert polynomials.

Instead of using equivariant classes associated to closed subvarieties of non-compact spaces the authors develop their theory in the context of multidegrees. The equivariant considerations for matrix Schubert varieties \([\overline X_w]\subseteq M_n\) are then done as multigraded commutative algebra for the Schubert determinantal ideals \(I_w\) cutting out the varieties \(\overline X_w\).

The Gröbner geometry of Schubert polynomials introduced provides a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. In fact the authors describe, for every matrix Schubert variety \(\overline X_w\);

(1) its multidegree and Hilbert series in terms of Schubert and Grothendieck polynomials

(2) a Gröbner basis consisting of minors in its defining ideal \(I_w\)

(3) the Stanley-Reisner complex \(\mathcal L_w\) of its initial ideal \(J_w\), which they prove is Cohen-Macaulay

(4) an inductive irredundant algorithm of weak Bruhat order for listing the facets of \(\mathcal L_w\).

The authors introduce a powerful inductive method that they call Bruhat induction, for working with determinantal ideals and their initial ideals. Bruhat induction as well as the derivation of the main theorems concerning Gröbner geometry rely on results concerning positivity of torus-equivariant cohomology classes represented by subschemes and shellability of certain simplicial complexes that reflect the nature of reduced subwords of words in Coxeter generators for Coxeter groups. The latter technique gives a new perspective, from simplicial topology, of the combinatorics of Schubert and Grothendieck polynomials.

Among the most important applications of the work is the geometrically positive formulae for Schubert polynomials, and connections with Fulton’s theory of degeneracy loci, relations between multidegrees and \(K\)-polynomials on \(n\times n\) matrices with classical cohomological theories on the flag manifold, and comparisons with the commutative algebra of determinantal ideals.

This article makes an important contribution to bridging the algebra and combinatorics of Schubert polynomials with the geometry of Schubert varieties. It brings new perspectives to problems in commutative algebra concerning ideals generated by minors of generic matrices, and provides a geometric context in which polynomial representatives for Schubert classes are uniquely singled out with no choices but a Borel subgroup of the general linear group \(\text{GL}_n\mathbb C\) in such a way that it is geometrically obvious that these representatives have nonnegative coefficients.

One of the main ideas in the article is to translate ordinary cohomological statements concerning Borel orbit closures on the flag manifold \(\mathcal Fl_n\) into equivariant-cohomological statements concerning double Borel orbit closures on the \(n\times n\) matrices \(M_n\). To be more precise, the preimage \(\widetilde X_w\subseteq \text{GL}_n\mathbb C\) of a Schubert variety \(X_w\in \mathcal Fl_n\) is an orbit closure for the action of the product \(B\times B^+\) of the lower and upper triangular subgroups of \(\text{GL}_n\mathbb C\) acting by multiplication on the left and right. When \(\overline X_w\subseteq M_n\) is the closure of \(\widetilde X_w\) and \(T\) is the torus in \(B\), the \(T\)-equivariant cohomology class \([\overline X_w]_T\in H_T^\ast(M_n)\) is the polynomial representative. It has positive coefficients because there is a \(T\)-equivariant flat (Gröbner) degeneration of \(\overline X_w\) to \(\mathcal L_w\) that is a union of coordinate subspaces \(L\subseteq M_n\). Each subspace \(L\subseteq \mathcal L_w\) has an equivariant cohomology class \([L]_T\in H_T^\ast(M_n)\) that is a monomial in \(x_1,\dots, x_n\), and the sum of these is \([\overline X_w]_T\). The formula is \([\overline X_w]_T = [\mathcal L_w]_T =\sum_{L\in \mathcal L_w}[L]_T\). More importantly, the authors identify a particularly natural degeneration of the matrix Schubert variety \(\overline X_w\) with a reduced and Cohen-Macaulay limit \(\mathcal L_w\) in which the subspaces have combinatorial interpretations and coincides with known combinatorial formulas for Schubert polynomials.

Instead of using equivariant classes associated to closed subvarieties of non-compact spaces the authors develop their theory in the context of multidegrees. The equivariant considerations for matrix Schubert varieties \([\overline X_w]\subseteq M_n\) are then done as multigraded commutative algebra for the Schubert determinantal ideals \(I_w\) cutting out the varieties \(\overline X_w\).

The Gröbner geometry of Schubert polynomials introduced provides a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. In fact the authors describe, for every matrix Schubert variety \(\overline X_w\);

(1) its multidegree and Hilbert series in terms of Schubert and Grothendieck polynomials

(2) a Gröbner basis consisting of minors in its defining ideal \(I_w\)

(3) the Stanley-Reisner complex \(\mathcal L_w\) of its initial ideal \(J_w\), which they prove is Cohen-Macaulay

(4) an inductive irredundant algorithm of weak Bruhat order for listing the facets of \(\mathcal L_w\).

The authors introduce a powerful inductive method that they call Bruhat induction, for working with determinantal ideals and their initial ideals. Bruhat induction as well as the derivation of the main theorems concerning Gröbner geometry rely on results concerning positivity of torus-equivariant cohomology classes represented by subschemes and shellability of certain simplicial complexes that reflect the nature of reduced subwords of words in Coxeter generators for Coxeter groups. The latter technique gives a new perspective, from simplicial topology, of the combinatorics of Schubert and Grothendieck polynomials.

Among the most important applications of the work is the geometrically positive formulae for Schubert polynomials, and connections with Fulton’s theory of degeneracy loci, relations between multidegrees and \(K\)-polynomials on \(n\times n\) matrices with classical cohomological theories on the flag manifold, and comparisons with the commutative algebra of determinantal ideals.

Reviewer: Dan Laksov (Stockholm)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14M12 | Determinantal varieties |

13C40 | Linkage, complete intersections and determinantal ideals |

14N15 | Classical problems, Schubert calculus |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

05E05 | Symmetric functions and generalizations |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

05E25 | Group actions on posets, etc. (MSC2000) |