Vertex operators, symmetric functions, and the spin group Γn

AbstractThis work provides a vertex operator approach to the symmetric group Sn and its double covering group Γn. By generalizing a result of Frenkel and Sato for Sn we formulate a correspondence between the space V̂ of certain twisted vertex operators, the ring Λ of symmetric functions over Q(√2), and the space of nontrivial irreducible characters of Γn. Under this identification we show that a distinguished orthogonal basis of V̂ corresponds to the set of nontrivial irreducible characters of Γn, where both are parametrized by partitions with odd integer parts. The counterpart of this distinguished basis in the ring Λ over Q(√2) is the set of Schur's Q-functions, which are, loosely speaking, the square roots of the Schur functions. The nontrivial part of the character table of Γn is shown to be given by certain matrix coefficients in V̂