Vanishing theorems and character formulas for the Hilbert scheme of points in the plane

By Mark D. Haiman. 2001

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Abstract.In an earlier paper [14], we showed that the Hilbert scheme of points in the plane Hn=Hilbn(ℂ2) can be identified with the Hilbert scheme of regular orbits ℂ2n//Sn. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author in [9]. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n+1)n-1.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5d96728df9e63f9032970e8f980120cae951a88

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Citations in the corpus

Lectures on instanton counting (2003, background): “) When ~τ 6= 0, it was studied in [34].”
Instanton counting on blowup. I. 4-dimensional pure gauge theory (2003, background): “n),E). Thus we can consider the associated Hilbert series chRiπ ∗E ≡ chHi(M(r,n),E). Now we can ﬁnish the proof of Proposition 4.1 thanks to a general result of Thomason [54]. The argument appears in [24] for r = 1, and his argument can be applied to our situation, once the above property of the coo”
Affine Springer fibers of type A and combinatorics of diagonal coinvariants (2012, methodology): “d y-degrees. Hence we can consider its bigraded Frobenius series F(DRn,z;q,t). The following theorem proved by using the geometry of Hilbert schemes expresses F(DRn,z;q,t) in terms of ∇. Theorem 1.1 ([10]). We have F(DRn,z;q,t) = ∇en(z). Here en(z) is the elementary symmetric function of degree n. A”
The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “In the case k D mn C 1 , the graded dimensions of conjecturally equivalent ﬁltrations can be readily calculated in a different geometric incarnation of the Cherednik algebra; see Haiman [39] and Gordon and Stafford [28; 29].”
FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, background): “In 2002 Haiman established a formula for the characters of the Sn–representations C(2)(n) in terms of MacDonald polynomials, and deduces a number of combinatorial consequences for the spaces C(2)(n) [Hai02b].”
Torus knots and the rational DAHA (2012, background): “) Sn is called the space of diagonal harmonics. It carries two natural gradings (x-degree andydegree) compatible with the action of Sn. The following results were conjectured in [Hai94] and proved in [Hai02a]: Theorem 6.11. The dimension of the space DHn equals (n+ 1)n−1. The dimension of its anti-i”
A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path (2010, background): “to form expressions for this character. At the root of this theory is a linear operator ∇, introduced in [BGHT99], under which the modiﬁed Macdonald polynomials H˜ µ[X;q,t] are eigenfunctions. Haiman [Hai02] proved that the Frobenius image of the character of DHn equals ∇en. This gives an explicit e”
A proof of the q,t-Schröder conjecture (2004, unknown): “We discuss how our formulas for 〈∇en, en−dhd〉 and 〈∇en, hn−dhd〉 prove two special cases of a recent conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov.”
Multivariate diagonal coinvariant spaces for complex reflection groups (2011, background): “To better see what is striking here, it is worth recalling that , although the case r = 1 has a long history [4, 13, 14], it is only recently that the special case r = 2 has been somewhat settled [3, 6, 7, 10].”