53 pages. This version is around half the length of v2, having been trimmed and completely reorganized. This is the final version, to appear in Annals of Mathematics. A substantial chunk of the cut material, on subword complexes, appears in its own paper (math.CO/0309259)
Our main theorems provide 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. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics.
Given a permutation w in S_n, we consider a determinantal ideal I_w 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 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 x n matrix;
- the Stanley-Reisner complex of the initial ideal in terms of known combinatorial diagrams associated to permutations in S_n; and
- a procedure inductive on weak Bruhat order for listing the facets of this complex, thereby generating the coefficients of Schubert polynomials by a positive recursion on combinatorial diagrams.
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.
Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gr{ö}bner bases for Tate algebras. We prove an analogue of the Buchberger criterion in our framework and design a Buchberger-like and a F4-like algorithm for computing Gr{ö}bner bases over Tate algebras. An implementation in SM is also discussed.
In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{ö}bner bases, even if the input is polynomials, the size of the output grows with the required precision, both in terms of the size of the coefficients and the size of the support of the series. We prove that ideals which are spanned by polynomials admit a Tate Gr{ö}bner basis made of polynomials, and we propose an algorithm, leveraging Mora's weak normal form algorithm, for computing it. As a result, the size of the output of this algorithm g…
We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansion of a truncated Schubert polynomial. Combining these results, we give a nonnegative combinatorial rule for the product of a Schubert and a Schur polynomial in the Schubert basis.
The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the $6$-vertex ice model (recently rediscovered by Lam, Lee, and S…
The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define parabolic analogues of quantum double Schubert polynomials, and show that they represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.
Enriched versions of type A Schubert polynomials are constructed with coefficients in a polynomial ring in variables $c_1, c_2, \ldots$. Specializing these variables to $0$ recovers the double Schubert polynomials of Lascoux and Schützenberger; specializing them to certain power series recovers the back-stable double Schubert polynomials of Lam, Lee, and Shimozono; specializing them to Schur Q-polynomials relates them to the type C double Schubert polynomials of Ikeda, Mihalcea, and Naruse. Many formulas for classical Schubert polynomials generalize to this setting. They give, and are characte…
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo-Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo-Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo-Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo-Mumfor…
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive t…
Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words of a permutation. In this paper, we give an explicit bijection between these two models, thereby definitively proving Kohnert's rule for Schubert polynomials.
We survey the recent study of involution Schubert polynomials and a modest generalization that we call degenerate involution Schubert polynomials. We cite several conditions when (degenerate) involution Schubert polynomials have simple factorization formulae. Such polynomials can be computed by traversing through chains in certain weak order posets, and we provide explicit descriptions of such chains in weak order for involutions and degenerate involutions. As an application, we give several examples of how certain multiplicity-free sums of Schubert polynomials factor completely into very simp…
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic Schubert calculus on these spaces. We also give a method to compute the natural arithmetic intersection numbers which arise, and prove that they are all rational numbers.
We show that the Schubert polynomial S_w specializes to the Catalan number C_n when $w=1(n+1)...2$. Several proofs of this result as well as a q-analog are given. An application to the singularities of Schubert varieties is given.
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize …
Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they are all rational numbers.
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other known bases including key polynomials and fundamental slide polynomials. We unify these and existing models by giving simple bijections between the combinatorial objects indexing each. In particular, we give a simple bijective proof that the balanced tableaux of Edelman and Greene enumerate reduced expressions and a direct combinatorial proof of Kohnert's algori…
We show that Lapointe-Lascoux-Morse k-Schur functions (at t=1) and Fomin-Gelfand-Postnikov quantum Schubert polynomials can be obtained from each other by a rational substitution. This is based upon Kostant's solution of the Toda lattice and Peterson's work on quantum Schubert calculus.
We show that the k-double Schur functions defined by the authors, and the quantum double Schubert polynomials studied by Kirillov and Maeno and by Ciocan-Fontanine and Fulton, can be obtained from each other by an explicit rational substitution. The main new ingredient is an explicit computation of Kostant's solution to the Toda lattice in terms of equivariant Schubert classes.
We give a new proof that three families of polynomials coincide: the double Schubert polynomials of Lascoux and Schützenberger defined by divided difference operators, the pipe dream polynomials of Bergeron and Billey, and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a "co-transition formula" which we explain to be some extent projectively dual to Lascoux' transition formula. We comment on the K-theoretic extensions.
We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus.
The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,...]. We suggest the "prism tableau model" for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Groebner geometry of matrix Schubert varieties.
Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety of a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for the Weyl groups of the classical types.
This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of the results of Can-Joyce-Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.
We examine the relationship between the (double) Schubert polynomials of Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl group invariants. We obtain generators for the kernel of the natural map from the corresponding ring of Schubert polynomials to the (equivariant) cohomology ring of symplectic and orthogonal flag manifolds.
We prove that twisted versions of Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$ are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the $\widetilde{\mathfrak S}_w$ as well as their localizations.
The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph. We define path Schubert polynomials, which are quantum cohomology analogues of skew Schubert polynomials recently introduced by Lenart and Sottile. They are given by sums over paths in the quant…
For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a fini…
In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.
Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar's nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on non-commutative k-Schur functions.
Ikeda-Mihalcea-Naruse's double Schubert polynomials represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined in terms of (flagged) marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-f…
The work of Buch and Fulton established a formula for a general kind of degeneracy locus associated to an oriented quiver of type $A$. The main ingredients in this formula are Schur determinants and certain integers, the quiver coefficients, which generalize the classical Littlewood-Richardson coefficients. Our aim in this paper is to prove a positive combinatorial formula for the quiver coefficients when the rank conditions defining the degeneracy locus are given by a permutation. In particular, this gives new expansions for Fulton's universal Schubert polynomials and the Schubert polynomials…
We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri's rule for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure …
We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.
In our previous work arXiv:1305.3543, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schutzenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of arXiv:0908.3628, which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elu…
We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-$A$ affine Weyl group. We construct two one-parameter families of functions that respectively transition positively with Hall-Littlewood and Macdonald's $P$-functions, and specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. Our approach leads us to conjecture that all elements in a defining …
Define a ``truncation'' $r_{t}(p)$ of a polynomial $p$ in $\{x_1,x_2,x_3,...\}$ as the polynomial with all but the first $t$ variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulae for certain Schubert structure constants in $K(Flags({\mathbb C}^n))$, in particular generalizing those from [Kogan '00] in which only cohomology was treated, and from [Buch `02] on the Grassmannian case. The terms of the answer are computed using ``marching'' ope…
We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types B, C, D which naturally extends the family of Lascoux of Schutzenberger in type A. These polynomials satisfy positivity, orthogonality, and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When w is a maximal Grassmannian element of the Weyl group, P_w(X,Y) can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type A formula of Kempf and Laksov. An example, motivated by quantum cohomology, shows that there are no Chern class…
Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are c…
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by …
The closure of a generic torus orbit in the flag variety $G/B$ of type $A$ is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincare polynomial of a generic torus orbit closure in a Schubert variety in $G/B$. When the generic torus orbit closure in a Schubert variety is smooth, its Poincare polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.
The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur funct…
We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws.
In this article, the comodule structure of Chow rings of Flag manifolds $\operatorname{CH}(G/B)$ is described by Schubert cells. Its equivariant version gives rise to a Hopf structure of the equivariant cohomology of flag manifolds $H^*_B(G/B)$. We get two identities of generalized Schubert polynomials as explanations of the geometric facts.
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert ba…
We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus.
Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.
We illuminate the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes. Equivalently, the structure constants for the ring of polynomials in variables $x_1,x_2,...$ in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas f…
