The Farahat-Higman ring of wreath products and Hilbert schemes

We study the structure constants of the class algebra $R_Z(G_n)$ of the wreath products $G_n$ associated to an arbitrary finite group G with respect to the basis of conjugacy classes. We show that a suitable filtration on $R_Z(G_n)$ gives rise to the graded ring $\mathcal G_G(n)$ with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring $\mathcal G_G$. The real conjugacy classes of G come to play a distinguished role, and is treated in detail in the case when G is a subgroup of $SL_2(C)$. The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface.Comment: latex, abstract/introduction modified, to appear in Advances in Mat

Equivariant K-theory, wreath products, and Heisenberg algebra

Given a finite group G and a G-space X, we show that a direct sum F_G (X) = \bigoplus_{n \geq 0}K_{G_n} (X^n) \bigotimes \C admits a natural graded Hopf algebra and $\lambda$-ring structure, where $G_n$ denotes the wreath product $G \sim S_n$. $F_G (X)$ is shown to be isomorphic to a certain supersymmetric product in terms of K_G(X)\bigotimes \C as a graded algebra. We further prove that $F_G (X)$ is isomorphic to the Fock space of an infinite dimensional Heisenberg (super)algebra. As one of several applications, we compute the orbifold Euler characteristic $e(X^n, G_n)$.Comment: 23 pages, some reorganizations and improvement of presentations, and other minor changes, to appear in Duke Math.

Classification of irreducible modules of W_3 algebra with c = -2

We construct irreducible modules V_{\alpha}, \alpha \in \C over W_3 algebra with c = -2 in terms of a free bosonic field. We prove that these modules exhaust all the irreducible modules of W_3 algebra with c = -2. Highest weights of modules V_{\alpha}, \alpha \in \C with respect to the full (two-dimensional) Cartan subalgebra of W_3 algebra are (\alpha(\alpha -1)/2, \alpha(\alpha -1)(2\alpha -1)/6). They are parametrized by points (t, w) on a rational curve w^2 - t^2 (8t + 1)/9 = 0. Irreducible modules of vertex algebra W_{1+\infty} with c = -1 are also classified.Comment: Latex, 22 page

Spin Hecke algebras of finite and affine types

We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are established.Comment: 24 pages, to appear in Adv. in Mat