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...

Let Λ be the space of symmetric functions and V k be the subspace spanned by the modified Schur functions {S λ [X/(1 − t)]} λ1≤k . We introduce a new family of symmetric polynomials, {A, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials A (k) λ [X; t] form a basis for V k and that the Macdonald polynomials indexed by partitions whose first part is not larger than k expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the A (k) λ [X; t] relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the A (k) λ [X; t] seem to play the same role for V k as the Schur functions do for Λ. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients.

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.

Abstract. We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a HallLittlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇e n [X]. We bring to light that certain generalized Hall-Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to su(ℓ) are shown to be k-Littlewood Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.