Vertex operators, symmetric functions, and the spin group Γn

Abstract

This work provides a vertex operator approach to the symmetric group S n and its double covering group Γ n . By generalizing a result of Frenkel and Sato for S n 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 Γ n . Under this identification we show that a distinguished orthogonal basis of V̂ corresponds to the set of nontrivial irreducible characters of Γ n , 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 Γ n is shown to be given by certain matrix coefficients in V̂.