Hilbert schemes, polygraphs and the Macdonald positivity conjecture

By Mark D. Haiman. 2000

559 citations recorded by the source corpus.

We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n is normal, Cohen-Macaulay, and Gorenstein, and hence flat over H_n. We derive two important consequences. (1) We prove the strong form of the "n! conjecture" of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K_{lambda,mu}(q,t). This establishes the Macdonald positivity conjecture, that K_{lambda,mu}(q,t) is always a polynomial with non-negative integer coefficients. (2) We show that the Hilbert scheme H_n is isomorphic to the Hilbert scheme of orbits C^2n//S_n, in such a way that X_n is identified with the universal family over C^2n//S_n.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7161f8caad41883dbb75bb758bb4e90446815d3f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. I. Gordon
2. E. Gorsky
3. I. Losev
4. Thomas Lam
5. V. Ginzburg
6. James Haglund
7. Pavel Etingof
8. Alexei Oblomkov

In Mathematical Physics

1. I. Losev
2. E. Gorsky
3. I. Gordon
4. V. Ginzburg
5. Pavel Etingof
6. Alexei Oblomkov
7. J. T. Stafford
8. Andrei Neguct

In Statistical and Nonlinear Physics

1. I. Gordon
2. J. T. Stafford
3. V. Ginzburg
4. Pavel Etingof
5. Thomas Lam
6. I. Losev
7. D. Arinkin
8. Sami H. Assaf

Citations in the corpus

• A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path (2010, background): “at (2.4) f[X+ z] = X k0 zk(h? k f)[X]; (2.5) f[X z] = X k0 ( z)k(e? k f)[X]: In addition we will refer to the form of the Macdonald basis H~ [X;q;t] that is relevant to the study of the n! Theorem [Hai01] and the q;t-Catalan numbers. The relation of this basis to the integral form J [X;q;t] of [Mac9”
• Rational Cherednik algebras and Hilbert schemes (2004, methodology): “is much more subtle as it is difficult to keep close control of the filtration on Bij . Our proof leans heavily on the work of Haiman [ Ha3 ,Ha4] surrounding the n! and polygraph theorems and the strategy is outlined in more detail in (6.6).”
• A Combinatorial Formula for Nonsymmetric Macdonald Polynomials (2008, methodology): “In fact, we gave two formulas, one for the transformed Macdonald polynomials H̃μ(x; q, t), which appear in the positivity theorem of [8] and are connected to the geometry of Hilbert schemes, and another [6, Proposition 8.”
• Affine Insertion and Pieri Rules for the Affine Grassmannian (2006, methodology): “The cyclic components of A are [9, 1], [3, 4] and [6, 6] and we have cA = (s1s0s9)(s4s3)(s6) = (s4s3)(s1s0s9)(s6) = · · · .”
• Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity (2010, methodology): “on of Kostka-Macdonald coefficients as the graded multiplicity of an irreducible representation in the Garsia-Haiman module, and hence Ke λ,µ(q,t) ∈N[q,t]. Following an idea outlined by Procesi, Haiman [Hai01] proved this conjecture by analyzing the algebraic geometry of the isospectral Hilbert scheme”
• A k-tableau characterization of k-Schur functions (2005, background): “involved the study of Macdonald polynomials. The Schur basis is again fundamental in this setting since the Macdonald expansion coefficients in this basis have a representation theoretic interpretation [6, 7, 11]. In work with Lascoux on the Macdonald polynomials [15], we discovered a new family of sy”
• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane (2001, methodology): “The canonical projection X → V induces a morphism X/G → V/G whose composite with the isomorphism V //G ∼= X/G yields the Chow morphism.”
• A combinatorial formula for non-symmetric Macdonald polynomials (2006, methodology): “a for Macdonald’s symmetric functions (for root systems of type A). In fact, we gave two formulas, one for the transformed Macdonald polynomials H˜ µ(x;q,t), which appear in the positivity theorem of [8] and are connected tothe geometry ofHilbert schemes, and another [6, Proposition 8.1]forthe integ”
• Representations of Rational Cherednik algebras with minimal support and torus knots (2013, methodology): “ e n−2jHen−2j for j≥0. Theorem 7.1 is proved in the next three subsections. 7.2. Procesi bundle. In the proof we use a remarkable bundle on X, the Procesi bundle Poriginally constructed by M. Haiman, [Hai]; an alternative construction was produced by Ginzburg, [Gi2]. The Hamiltonian reduction constr”
• Symplectic reflection algebras (2007, background): “[17, Theorem 1.1], produces such as equivalence, but it is rather implicit and we require something more explicit: for Γ = {1}this is essentially a combination of the celebrated n! theorem of Haiman, [89], and the celebrated derived McKay correspondence of Bridgeland–King–Reid, [21]; the case of gen”
• Torus knots and the rational DAHA (2012, background): “02a]). The following isomorphisms hold: 26 E. GORSKY, A. OBLOMKOV, J. RASMUSSEN, AND V. SHENDE (1) Hi(Z n,P⊗Tk) = 0 for all k,nand i>0. (2) DHn = H0(Zn,P⊗T). (3) (DHn)ε = H0(Zn,ΛnT). Theorem 6.14 ([Hai01]). Let Iµ denote the monomial ideal in C[x,y]correspondingto a Young diagram µ. Let Jµ denote”
• A note on automorphisms and birational transformations of holomorphic symplectic manifolds (2009, background): “Let Zn be the isospectral Hilbert scheme, which is defined by Haiman [16] as the reduced fiber product of S with S over S and Z̃n → Zn a resolution of singularities.”
• Almost-commuting variety, D-modules, and Cherednik algebras (2004, background): “In an earlier paper, Haiman showed, cf. [Ha1, Proposition 2.13], that the above theorem holds for all k ≫ 0. The corresponding statement for all k follows from Haiman’s proof of his his Polygraph theorem, the main technical result in [ Ha2 ].”