Proposes comet-shaped quiver gauge theories for surface defects with nested instantons in 4D gauge theories on T^2 × T*C_{g,k} and gives conjectural explicit formulae for the virtual equivariant elliptic genus of bundles over nested Hilbert schemes of points on the affine plane.
Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
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abstract
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.
verdicts
UNVERDICTED 2representative citing papers
Derives the tangent space as a compatibility kernel and a modified arm-leg weight formula at torus-fixed points of (A^2)^[n,n+1] indexed by partitions with removable corners, with Macaulay2 checks up to size 16.
citing papers explorer
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Defects, nested instantons and comet shaped quivers
Proposes comet-shaped quiver gauge theories for surface defects with nested instantons in 4D gauge theories on T^2 × T*C_{g,k} and gives conjectural explicit formulae for the virtual equivariant elliptic genus of bundles over nested Hilbert schemes of points on the affine plane.
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Deformation Theory and Torus-Fixed Geometry of the Nested Hilbert Scheme of Points
Derives the tangent space as a compatibility kernel and a modified arm-leg weight formula at torus-fixed points of (A^2)^[n,n+1] indexed by partitions with removable corners, with Macaulay2 checks up to size 16.