We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. 2) A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of the pairing between homology and cohomology of the affine Grassmannian.
These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously.
Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.
We give a Pieri rule for the torus-equivariant cohomology of (submaximal) Grassmannians of Lie types B, C, and D. To the authors' best knowledge, our rule is the first manifestly positive formula, beyond the equivariant Chevalley formula. We also give a simple proof of the equivariant Pieri rule for the ordinary (type A) Grassmannian.
We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent work of Thomas and Yong and of E. Clifford has shown that the full Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K-theoretic Pieri coefficients both as i…
The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li a…
An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL_n and a similar formula is conjectured for Sp_{2n} and SO_{2n+1}. For SL_n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varie…
We provide a generalization of the Schensted insertion algorithm for rc-graphs of Bergeron and Billey. The new algorithm is used to give a new proof of Pieri's formula.
We give quantum Pieri rules for quantum cohomology of Grassmannians of classical types, expressing the quantum product of Chern classes of the tautological subbundles with general cohomology classes. We derive them by showing the relevant genus zero, three-pointed Gromov-Witten invariants coincide with certain classical intersection numbers.
We give elementary and short proofs of the Littlewood-Richardson rules for type A Grassmannians and maximal isotropic Grassmannians, based on the corresponding Pieri rules.
We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form (k-p+1,k-p+2,...,k+1) or the form (k+p,k+p-1,...,k), and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the k-Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in …
The Pieri rule is a nonnegative, multiplicity-free formula for the Schur function expansion of the product of an arbitrary Schur function with a single row Schur function. Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur function basis of symmetric functions to a basis of the full polynomial ring. We prove a nonsymmetric generalization of the Pieri rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a single part key polynomial. Our proof is com…
We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and k-Schur functions.
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.
We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.
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 …
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 give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group ${\mathcal B…
Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schu…
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.
We describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the…
We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case after the (co)minuscule family (where a rule has been proved by H.Thomas and the second author using work of R.Proctor). Our results build on earlier Pieri-type rules of P.Pragacz-J.Ratajski and of A.Buch-A.Kresch-H.Tamvakis. Specifically, our formula for OG(2,2n) is the first complete rule for a case where diagrams are non-planar. Yet the formulas possess …
