Correlation Functions of Strict Partitions and Twisted Fock Spaces

Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch-Okounkov [BO], and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions. We find closed formulas for the 1-point functions in both cases in terms of Jacobi θ-functions. These correlation functions afford several distinct interpretations. Introduction Bloch-Okounkov [BO] studied certain correlation functions on the infinite wedge representation (i.e. the Fock space of a pair of free fermions), which is closely related to the correlation functions of vertex operators [Zhu]. Recently it has gradually become clear that this correlation function (and its variant) affords several distinct interpretations: (1) It can be regarded as a certain character of representations of the W1+∞ algebra of level one. (2) It can be regarded as the q-expectation value of a certain function on the set of partitions. (3) It can be interpreted as a generating function of the stationary GromovWitten invariants of an elliptic curve. (4) A variant of this correlation function can be interpreted as a generating function of equivariant intersection numbers on Hilbert schemes of points on the affine plane. (5) It can be interpreted as a generating function of certain structure constants of the class algebras of the symmetric groups. The items (1) and (2) were points of view taken in [BO] (also see [Ok]). More on representation theory of W1+∞ (which is a central extension of the Lie algebra of differential operators on the circle) of positive integral levels using Fock spaces can be found in [FKRW, AFMO]. Item (3) is due to the recent remarkable work of Okounkov-Pandharipande [OP]. Item (4) is due to Li, Qin, and the author [LQW], and (5) is essentially equivalent to (4) based on the results of Vasserot, Lascoux-Thibon and ours (cf. [LQW]; also see [LT, Wa]). We refer to [LT, Wa] for a closely related study of the class algebras of the symmetric groups and more generally of wreath products and their connections to the W1+∞ algebra. Partially supported by an NSF grant. 1