Peterson varieties are special nilpotent Hessenberg varieties that have appeared in the study of quantum cohomology, representation theory, and combinatorics. In type $A$, the Peterson variety $Y$ is a subvariety of the complete flag variety $Fl(n; \mathbb C)$, and is invariant under the action of a subgroup $S\cong \mathbb C^*$ of $T$, where $T$ is the standard (noncompact) torus acting on $Fl(n; \mathbb C)$. Using the Peterson Schubert basis introduced by Harada and Tymoczko obtained by restricting a specific set of Schubert classes from $H_T^*(Fl(n; \mathbb C))$ to $H_S^*(Y)$, we describe the product structure of the equivariant cohomology $H_{S}^*(Y)$. In particular, we show that the product is manifestly positive in an appropriate sense by providing an explicit positive combinatorial formula for its structure constants. Our method requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde's identity.
Let $Φ$ be a root system. Postnikov introduced and studied the mixed $Φ$-Eulerian numbers. These numbers indicate the mixed volumes of $Φ$-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget-Spink-Tseng gave a simple computation for the mixed $Φ$-Eulerian numbers when $Φ$ is of type $A$. In this paper we connect a relation between mixed $Φ$-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget-Sp…
The main result of this note is a Giambelli formula for the Peterson Schubert classes in the $S^1$-equivariant cohomology ring of a type $A$ Peterson variety. Our results depend on the Monk formula for the equivariant structure constants for the Peterson Schubert classes derived by Harada and Tymoczko. In addition, we give proofs of two facts observed by H. Naruse: firstly, that some constants which appear in the multiplicative structure of the $S^1$-equivariant cohomology of Peterson varieties are Stirling numbers of the second kind, and secondly, that the Peterson Schubert classes satisfy a …
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculu…
We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form o…
We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and structure constants. This generalizes Peterson's isomorphism in (co)homology. We prove a formula for the Pontryagin structure constants in the K-homology ring, and we use it to check our conjecture in few situations.
This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explic…
We prove two lemmata about Schubert calculus on generalized flag manifolds G/B, and in the case of the ordinary flag manifold GL_n/B we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen_descent-cycling_. Computer experiment shows that these lemmata suffice to determine all of GL_n Schubert calculus through n=5, and 99.97%+ at n=6. We use them to give a quick proof of Monk's rule. The lemmata also hold in equivariant (``double'') Schubert calculus for Kac-Moody groups G.
We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.
Let G be a simple and simply-connected complex algebraic group, and let X \subset G^\vee be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P. The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity…
We investigate double transitivity of Galois groups in the classical Schubert calculus on Grassmannians. We show that all Schubert problems on Grassmannians of 2- and 3-planes have doubly transitive Galois groups, as do all Schubert problems involving only special Schubert conditions. We use these results to give a new proof that Schubert problems on Grassmannians of 2-planes have Galois groups that contain the alternating group. We also investigate the Galois group of every Schubert problem on Gr(4,8), finding that each Galois group either contains the alternating group or is an imprimitive p…
We give a nonrecursive, combinatorial characterization of multiplicity-free products of Grassmannian Schubert classes. This answers a question of W. Fulton and extends results of J. Stembridge.
The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author.
The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming from the Chern character.
This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus
Let G be a semi-simple algebraic group over ${\mathbb C}$, B a Borel subgroup of G and T a maximal torus in B. A beautiful unpublished result of Dale Peterson says that if G is simply laced, then every rationally smooth point of a Schubert variety X in G/B is nonsingular in X. The purpose of this paper is to generalize this result to arbitrary T-stable subvarieties of G/B, the only restriction being that G contains no $G_2$ factors. In particular, we show that a Schubert variety X in such a G/B is nonsingular if and only if all the reduced tangent cones of X are linear.
We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.
We continue the study of root-theoretic Young diagrams (RYDs) from [Searles-Yong '13]. We provide an RYD formula for the $GL_n$ Belkale-Kumar product, after [Knutson-Purbhoo '11], and we give a translation of the indexing set of [Buch-Kresch-Tamvakis '09] for Schubert varieties of non-maximal isotropic Grassmannians into RYDs. We then use this translation to prove that the RYD formulas of [Searles-Yong '13] for Schubert calculus of the classical (co)adjoint varieties agree with the Pieri rules of [Buch-Kresch-Tamvakis '09], which were needed in the proofs of the (co)adjoint formulas.
We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'' and the second one is written in terms of factorial Schur $Q$- or $P$-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.
Fulton and Woodward have recently identified the smallest degree of $q$ that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result, and provide an alternative characterization of this smallest degree in terms of the rim hook formula for the quantum product.
We describe recent work on positive descriptions of the structure constants of the cohomology of homogeneous spaces such as the Grassmannian, by degenerations and related methods. We give various extensions of these rules, some new and conjectural, to K-theory, equivariant cohomology, equivariant K-theory, and quantum cohomology.
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 …
We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.
In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, f…
The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes who…
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 describe a relationship between work of Laksov, Gatto, and their collaborators on realizations of (generalized) Schubert calculus of Grassmannians, and the geometric Satake correspondence of Lusztig, Ginzburg, and Mirković and Vilonen. Along the way we obtain new proofs of equivariant Giambelli formulas for the ordinary and orthogonal Grassmannians, as well as a simple derivation of the "rim-hook" rule for computing in the equivariant quantum cohomology of the Grassmannian.
This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson-W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.
The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of m-planes in complex n-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative k-Schur functions. As an application, we …
We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Schützenberger '77] for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of [Buch '02] and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending [Thomas-Yong '06]. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger.
One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this paper, we explicitly describe string parametrizations of opposite Demazure crystals, which give a natural generalization of reduced dual Kogan faces. We also relate reduced Kogan faces with Demazure crystals using the theory of …
We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Peterson's quantum version of Chevalley's formula, also for general G/P's.
Patch ideals encode neighbourhoods of a variety in GL_n/B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen-Macaulay and Gorenstein. Consequently, we combinatorially describe the singular locus of the Peterson variety; give an explicit equivariant K-theory localization formula; and extend some results of [B. Kostant '96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties. We conjecture that the projectivized tangent cones are Cohen-Macaulay, and that their h-polynomials are nonnegative and up…
Following Contou-Carrere [CC], we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. We prove that the rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called, the combinatorial tangent space. For this, we use a characterization of rational smoothness of a Schubert variety introduced by Carrell and Peterson [CP].
A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey-Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson's theorem that all simply-laced…
We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case G = SL_n, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called K-k-Schur functions, whose highest degree term is a k-Schur function. The dual basis in K-cohomology is given …
Traditional formulations of geometric problems from the Schubert calculus, either in Plucker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and (in local coordinates) typically of degree exceeding two. We present an alternative primal-dual formulation using parametrizations of Schubert cells in the dual Grassmannians in which intersections of Schubert varieties become complete intersections of bilinear equations. This formulation enables the numerical certification of problems in the Schubert calcu…
We show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert cells defined by such flags meet transversally. This strengthens a result of Belkale and Kumar in arXiv:0708.0398.
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation. A similarly rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation. This showcases new possibilities for the use of computers in mathematical research.
In this paper we classify the multiplicity-free skew characters of the symmetric group. Furthermore we show that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the `inverse' partitions (Theorem 4.2).
We show a Z^2-filtered algebraic structure and a "quantum to classical" principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for any partial flag variety of Lie type A.
For any complex reductive connected Lie group G, many of the structure constants of the ordinary cohomology ring H^*(G/B; Z) vanish in the Schubert basis, and the rest are strictly positive. We present a combinatorial game, the ``root game'', which provides some criteria for determining which of the Schubert intersection numbers vanish. The definition of the root game is manifestly invariant under automorphisms of G, and under permutations of the classes intersected. Although these criteria are not proven to cover all cases, in practice they work very well, giving a complete answer to the ques…
We present a proof of a Littlewood-Richardson rule for the K-theory of odd orthogonal Grassmannians OG(n,2n+1), as conjectured in [Thomas-Yong '09]. Specifically, we prove that rectification using the jeu de taquin for increasing shifted tableaux introduced there, is well-defined and gives rise to an associative product. Recently, [Buch-Ravikumar '09] proved a Pieri rule for OG(n,2n+1) that [Feigenbaum-Sergel '09] showed confirms a special case of the conjecture. Together, these results imply the aforementioned conjecture.
In 1990, Lakshmibai and Sandhya published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern avoidance was used to characterize geometrical properties of Schubert varieties. Their results are very closely related to work of Haiman, Ryan and Wolper, but Lakshmibai-Sandhya were the first to use that language exactly. Pattern avoidance in permutations was used historically by Knuth, Pratt, Tarjan, and others in the 1960's and 1970's to characterize sorting algorithms in computer science. Lascoux and Sch$\text{…
We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.
We compute the Nash blow-up of a cominuscule Schubert variety. In particular, we show that the Nash blow-up is algebraically isomorphic to another Schubert variety of the same Lie type. As a consequence, we give a new characterization of the smooth locus and, for Grassmannian Schubert varieties, determine when the Nash blow-up is a resolution of singularities. We also study the induced torus action on the Nash blow-up and give a bijection between its torus fixed points and Peterson translates on the Schubert variety.
