We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials K λR (q) labeled by a partition λ and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible GL(n, C)-module V λ of highest weight λ in the tensor product V R 1 ⊗ · · · ⊗ V R L . (2000). Primary 05A19, 05A15. Mathematics Subject Classification

The symmetric group Sn acts on the polynomial ring Q[xn] = Q[x1, . . . , xn] by variable permutation. The invariant ideal In is the ideal generated by all Sn-invariant polynomials with vanishing constant term. The quotient Rn = Q[xn]In is called the coinvariant algebra. The coinvariant algebra Rn has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization I n,k ⊆ Q[xn] of the ideal In indexed by two positive integers k ≤ n. The corresponding quotient R n,k := Q[xn] I n,k carries a graded action of Sn and specializes to Rn when k = n. We generalize many of the nice properties of Rn to R n,k . In particular, we describe the Hilbert series of R n,k , give extensions of the Artin and Garsia-Stanton monomial bases of Rn to R n,k , determine the reduced Gröbner basis for I n,k with respect to the lexicographic monomial order, and describe the graded Frobenius series of R n,k . Just as the combinatorics of Rn are controlled by permutations in Sn, we will show that the combinatorics of R n,k are controlled by ordered set partitions of {1, 2, . . . , n} with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of R n,k is (up to a minor twist) the t = 0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded Sn-module V n,k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module R n,k solves this problem in the specialization t = 0.

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH * (G/P ) of a flag variety is, up to localization, a quotient of the homology H * (Gr G ) of the affine Grassmannian Gr G of G. As a consequence, all three-point genus zero Gromov-Witten invariants of G/P are identified with homology Schubert structure constants of H * (Gr G ), establishing the equivalence of the quantum and homology affine Schubert calculi.For the case G = B, we use the Mihalcea's equivariant quantum Chevalley formula for QH * (G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology.Our main results extend to the torus-equivariant setting.

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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...

This is a combinatorial study of the Poincaré polynomials of isotypic components of a natural family of graded G L(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka-Foulkes polynomials and are q-analogues of Littlewood-Richardson coefficients. The coefficients of two-column Macdonald-Kostka polynomials also occur as a special case. It is conjectured that these q-analogues are the generating function of so-called catabolizable tableaux with the charge statistic of Lascoux and Schützenberger. A general approach for a proof is given, and is completed in certain special cases including the Kostka-Foulkes case. Catabolizable tableaux are used to prove a characterization of Lascoux and Schützenberger for the image of the tableaux of a given content under the standardization map that preserves the cyclage poset.

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1, and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula was conjectured for by Buch and Fulton in terms of factor sequences of Young tableaux [BF99]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in "doubled" versions, two for double quiver polynomials, and the other two for their stable versions, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case.Our method begins by identifying quiver polynomials as multidegrees [Jos84, Ros89] via equivariant Chow groups [EG98]. Then we make use of Zelevinsky's map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are schemetheoretic isomorphisms (originally proved in [LM98] using standard monomial theory). Writing double Schubert polynomials in terms of pipe dreams [FK96] then provides another geometric formula for double quiver polynomials, via [KM03a]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for double quiver functions in terms of products of Stanley symmetric functions [Sta84]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92,KM03a] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a, RS98], and from there to a 'double stable' generalization of the Buch-Fulton conjecture.

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