Vertex operators, symmetric functions, and the spin group \(\Gamma_ n\)
This work generalizes, to the spin group of the symmetric group, the known mathematical correspondence between representation theory of the latter, and vertex operators on the symmetric algebra generated by elements of a Heisenberg algebra. For the spin group \(\Gamma_ n\) (the double covering of the symmetric group \(S_ n)\) the construction involves ``twisted'' vertex operators Y(z) acting on the completion \(\tilde V\{\) z,z\({}^{-1}\}\) of the Fock space generated by symmetric polynomials in the odd-moded multiplication operators of a Heisenberg algebra: \[ Y(z)=\exp (\sum^{\infty}_{n=1,\quad n\in Z_ 1}\frac{2z^ n}{n}h_{-n})\exp (-\sum^{\infty}_{n=1,\quad n\in Z_ 1}\frac{2z^{-n}}{n}h_ n). \] The key result (theorem 5.4) is the proof of an isometric isomorphism, under which an element \(Y_{-\lambda_ 1}Y_{-\lambda_ 2}...Y_{- \lambda_ k}1\) of \(\tilde V,\) derived from homogeneous components of Y(z) is mapped to the Schur Q-function associated with the partition \((\lambda_ 1,...,\lambda_ k)\), \(\lambda_ 1\geq...\geq \lambda_ k\). From this follow explicitly the properties of the Q-functions and concrete formulae for their role as irreducible characters in the representation theory of \(\Gamma_ n\).
scientific article; zbMATH DE number 4200466
scientific article