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Symmetric groups and the cup product on the cohomology of Hilbert schemes. (English) Zbl 1093.14008

Consider the Hilbert scheme \(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}})\) of generalized \(n\)-tuples of points on the affine plane: \(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}})\) is a smooth quasi-projective manifold of complex dimension \(2n\) and is a natural resolution of the quotient \((\mathbb{A}^2_{\mathbb{C}})^n/S_n\). Its odd cohomology vanishes, and there is no torsion. In this paper, the authors describe a natural model for the cohomology ring \(H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Z})\) in terms of a graded version of the ring \(\mathcal{C}(S_n)\) of integer-valued class functions on the symmetric group \(S_n\).
The model. Starting from the group ring \(\mathbb{Z}[S_n]\), define a degree by setting \(\text{deg}(\pi)=d\) if the permutation \(\pi\in S_n\) can be written as a minimal product of \(d\) transpositions. The product in \(\mathbb{Z}[S_n]\) is compatible with the increasing filtration associated to the degree, and induces a product on the graded space \(\text{gr}\mathbb{Z}[S_n]\). The subring \(\mathcal{C}(S_n)\subset \mathbb{Z}[S_n]\) inherits a structure of commutative graded ring.
The theorem. The graded rings \(H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Z})\) and \(\mathcal{C}(S_n)\) are naturally isomorphic (Theorem 1.1).
The proof. First, the authors prove the isomorphism over \(\mathbb{Q}\) by considering all values of \(n\) together and by induction on both \(n\) and the cohomological degree (Proposition 5.3). This uses the identification of \(\mathcal{C}:=\bigoplus_n\mathcal{C}(S_n)\otimes \mathbb{Q}\) and \(\mathbb{H}=\bigoplus_n H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Q})\) with the bosonic Fock space \(\mathcal{P}=\mathbb{Q}[p_1,p_2,\ldots]\). The vertex algebra isomorphisms \(\mathcal{C}\cong \mathcal{P}\) and \(\mathcal{P}\cong \mathbb{H}\) come respectively from I. B. Frenkel, W. Wang [J. Algebra 242, No. 2, 656–671 (2001; Zbl 0981.17021)] and H. Nakajima [Ann. Math. (2) 145, No. 2, 379–388 (1997; Zbl 0915.14001)]. Then the assertion results from the similarity of Goulden’s operator on \(\mathcal{C}\) with Lehn’s operator on \(\mathbb{H}\) (see Theorem 4.1 and M.Lehn [Invent. Math. 136, No. 1, 157–207 (1999; Zbl 0919.14001)]). In a second part, the authors show that the ring isomorphism \(\mathcal{C}(S_n)\otimes \mathbb{Q}\cong H^*(\text{Hilb}^n(\mathbb{A}^2_{\mathbb{C}});\mathbb{Q})\) also holds over the integers by exhibiting appropriate \(\mathbb{Z}\)-basis of both spaces (§6).
Remarks. 1) This result has been independently obtained by E. Vasserot [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7–12 (2001; Zbl 0991.14001)] using other methods.
2) The isomorphism has been further interpreted in the context of the Chen-Ruan conjecture by B. Fantechi, L. Göttsche [Duke Math. J. 117, No. 2, 197–227 (2003; Zbl 1086.14046)].
3) Generalizing this result, a similar description of the complex cohomology ring of any symplectic resolution of the quotient of a vector space by a finite group of symplectic automorphisms has been obtained later by V. Ginzburg and D. Kaledin [Adv. Math. 186, No. 1, 1–57 (2004; Zbl 1062.53074)].

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives
20B30 Symmetric groups
17B68 Virasoro and related algebras
17B69 Vertex operators; vertex operator algebras and related structures
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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References:

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