Affine insertion and Pieri rules for the affine Grassmannian

Abstract

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n, C). Our main results are:• 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.• A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H * (Gr).• A combinatorial interpretation of the pairing H * (Gr) × H * (Gr) → Z induced by the cap product. 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. Introduction vii Chapter 1. Schubert Bases of Gr and Symmetric Functions 1.1. Symmetric functions 1.2. Schubert bases of Gr 1.3. Schubert basis of the affine flag variety Chapter 2. Strong Tableaux 2.1. Sn as a Coxeter group 2.2. Fixing a maximal parabolic 2.3. Strong order and strong tableaux 2.4. Strong Schur functions Chapter 3. Weak Tableaux 3.1. Cyclically decreasing permutations and weak tableaux 3.2. Weak Schur functions 3.3. Properties of weak strips 3.4. Commutation of weak strips and strong covers Chapter 4. Affine Insertion and Affine Pieri 4.1. The local rule φ u,v 4.2. The affine insertion bijection Φ u,v 4.3. Pieri rules for the affine Grassmannian 4.4. Conjectured Pieri rule for the affine flag variety 4.5. Geometric interpretation of strong Schur functions Chapter 5. The Local Rule φ u,v 5.1. Internal insertion at a marked strong cover 5.2. Definition of φ u,v 5.3. Proofs for the local rule Chapter 6. Reverse Local Rule 6.1. Reverse insertion at a cover 6.2. The reverse local rule 6.3. Proofs for the reverse insertion Chapter 7. Bijectivity 7.1. External insertion v vi CONTENTS 7.2. Case A (commuting case) 7.3. Case B (bumping case): 7.4. Case C (replacement bump) Chapter 8. Grassmannian Elements, Cores, and Bounded Partitions 8.1. Translation elements 8.2. The action of Sn on partitions 8.3. Cores and the coroot lattice 8.4. Grassmannian elements and the coroot lattice 8.5. Bijection from cores to bounded partitions 8.6. k-conjugate 8.7. From Grassmannian elements to bounded partitions Chapter 9. Strong and Weak Tableaux Using Cores 9.1. Weak tableaux on cores are k-tableaux 9.2. Strong tableaux on cores 9.3. Monomial expansion of t-dependent k-Schur functions 9.4. Enumeration of standard strong and weak tableaux Chapter 10. Affine Insertion in Terms of Cores 10.1. Internal insertion for cores 10.2. E...

Schubert polynomials for the affine Grassmannian of the symplectic group

“…The Schubert basis of H * (Gr SLn(C) ) are the dual k-Schur functions [19] which are generalized by the affine Stanley symmetric functions [14]. In [16] Pieri rules were given for the multiplication of Bott's generators on the Schubert bases of Bott's realization of H * (Gr SLn(C) ) and H * (Gr SLn(C) ). Furthermore, a combinatorial interpretation of the pairing between H * (Gr SLn(C) ) and H * (Gr SLn(C) ) is given.…”

“…Pieri rule for H * (Gr Sp2n(C) ) and explicit description of homology Schubert basis. We hope to describe the symmetric functions {P (n) w | w ∈C 0 n } ⊂ Γ (n) explicitly in the future, perhaps in a manner similar to the strong tableaux in [16]. As is explained in [16], the description of P (n) w is essentially equivalent to the description of a Pieri rule for H * (Gr Sp2n(C) ).…”

Affine dual equivalence and $k$-Schur functions

“…At this time, a number of conjecturally equivalent definitions for k-Schur functions exist [17,18,19,20,21,22], making the term "k-Schur function" rather ambiguous. In this paper, we advocate for the geometrically inspired definition as the weighted generating function of starred strong tableaux presented by Lam, Lapointe, Morse and Shimozono [17].…”

“…In this paper, we advocate for the geometrically inspired definition as the weighted generating function of starred strong tableaux presented by Lam, Lapointe, Morse and Shimozono [17]. This definition at t = 1 is equivalent to the k-tableaux characterization in [22] which has been shown to represent the Schubert basis in the homology of the affine Grassmannian of type A [15].…”

“…In particular, we review an interesting order preserving bijection between a quotient of the affine symmetric group with the n-core partitions relating Bruhat order to a subposet of Young's lattice. In Section 3, one definition of k-Schur functions expanded into fundamental quasisymmetric functions is given following [17,Conjecture 9.11]. These functions can be indexed by n-cores, minimal length coset representatives for S n /S n , or k = n − 1 bounded partitions since all three sets are in bijection.…”