Virasoro algebra and wreath product convolution

We present a group theoretic construction of the Virasoro algebra in the framework of wreath products. This can be regarded as a counterpart of a geometric construction of Lehn in the theory of Hilbert schemes of points on a surface. Introduction It is by now well known that a direct sum ⊕ n≥0R(Sn) of the Grothendieck rings of symmetric groups Sn can be identified with the Fock space of the Heisenberg algebra of rank one. One can construct vertex operators whose components generate an infinite-dimensional Clifford algebra, the relation known as boson-fermion correspondence [F] (also see [J]). A natural open problem which arises here is to understand the group theoretic meaning of more general vertex operators in a vertex algebra [B, FLM]. A connection between a direct sum RΓ = ⊕ n≥0R(Γn) of the Grothendieck rings of wreath products Γn = Γ ∼ Sn associated to a finite group Γ and vertex operators has been realized recently in [W] and [FJW]. When Γ is trivial one recovers the above symmetric group picture. On the other hand, this wreath product approach turns out to be very much parallel to the development in the theory of Hilbert schemes of points on a surface (cf. [W, N2] and references therein). As one expects that new insight in one theory will shed new light on the other, this refreshes our hope of understanding the group theoretic meaning of general vertex operators. The goal of this paper is to take the next step in this direction to produce the Virasoro algebra within the framework of wreath products. Denote by Γ∗ the set of conjugacy classes of Γ, and by c the identity conjugacy class. Recall [M, Z] that the conjugacy classes of the wreath product Γn are parameterized by the partition-valued functions on Γ∗ of length n (also see Section 2). Given c ∈ Γ∗, we denote by λc the function which maps c to the one-part partition (2), c 0 to the partition (1) and other conjugacy classes to 0. We will define an operator ∆c in terms of the convolution with the characteristic class function on Γn (for all n) associated to the conjugacy class parameterized by λc. We show that this 1 2 Igor Frenkel and Weiqiang Wang operator can be identified with a differential operator which is the zero mode of a certain vertex operator when we identify RΓ as in [M] with a symmetric algebra. A group theoretic construction of Heisenberg algebra has been given in [W] (also see [FJW]) which acts on RΓ irreducibly. The commutator between ∆c and the Heisenberg algebra generators on RΓ provides us the Virasoro algebra generators. Our construction is motivated in part by the work of Lehn [L] in the theory of Hilbert schemes. Among other results, he showed that an operator defined in terms of intersection with the boundary of Hilbert schemes may be used to produce the Virasoro algebra when combined with earlier construction of Heisenberg algebra due to Nakajima and independently Grojnowski [N1, Gr]. It remains an important open problem to establish a precise relationship between Lehn’s construction and ours. We remark that the convolution operator in the symmetric group case (i.e. Γ trivial), when interpreted as an operator on the space of symmetric functions, is intimately related to the Hamiltonian in Categero-Sutherland integrable system and to the Macdonald operator which is used to define Macdonald polynomials [AMOS, M]. After we discovered our group theoretic approach toward the Virasoro algebra, we notice that our convolution operator in the symmetric group case has been considered by Goulden [Go] when studying the number of ways of writing permutations in a given conjugacy class as products of transpositions. We regard this as a confirmation of our belief that the connections between RΓ and (general) vertex operators are profound. It is likely that often time when we understand something deeper in this direction, we may realize that it is already hidden in the vast literature of combinatorics, particularly on symmetric groups and symmetric functions for totally different needs. Then the virtue of our point of view will be to serve as a unifying principle which patches together many pieces of mathematics which were not suspected to be related at all. The plan of this paper is as follows. In Sect. 1, we recall some basics of Heisenberg and Virasoro algebras from the viewpoint of vertex algebras. In Sect. 2 we set up the background in the theory of wreath products which our main constructions are based on. In Sect. 3 we present our main results. 1 Basics of Heisenberg and Virasoro algebras In the following we will present some basic constructions in vertex algebras which give us the Virasoro algebra from a Heisenberg algebra. Let L be a rank N lattice endowed with an integral non-degenerate symmetric bilinear form. Indeed we will only need the case L = Z with the standard bilinear form. Denote by h = C ⊗ Z L the vector space generated by L with the bilinear from Virasoro algebra and wreath product convolution 3 〈−,−〉 induced from L. We define the Heisenberg algebra ĥ = C[t, t] ⊗ h ⊕ CC with the following commutation relations: [C, an] = 0, [an, bm] = nδn,−m〈a, b〉C, where an denotes t n ⊗ a, a ∈ h. We denote by SL the symmetric algebra generated by ĥ − = tC[t] ⊗ h. It is well known that SL can be given the structure of an irreducible module over Heisenberg algebra ĥ by letting a−n, n > 0 acts as multiplication and letting an.a 1 −n1 a2−n2 . . . a k −nk = k ∑ i=1 δn,ni〈a, a i〉a1−n1a 2 −n2 . . . ǎi−ni . . . a k −nk , where n ≥ 0, ni > 0, a, a i ∈ h for i = 1, . . . , k, and ǎi−ni means the very term is deleted. A natural gradation on SL is defined by letting deg(a1−n1a 2 −n2 . . . a k −nk ) = n1 + . . .+ nk. We say an operator on SL is of degree p if it maps any n-th graded subspace of SL to (n + p)-th graded subspace. The space SL carries a natural structure of a vertex algebra [B, FLM]. It is convenient to use the generating function in a variable z: a(z) = ∑