Vertex operators, symmetric functions, and the spin group \(\Gamma_ n\)
type: Item
instance of: scholarly article
title: Vertex operators, symmetric functions, and the spin group \(\Gamma_ n\)
zbMATH Open document ID: 0727.20011
DOI: 10.1016/0021-8693(91)90177-A
author: Naihuan Jing
published in: Journal of Algebra
publication date: 1991-01-01T00:00:00Z
review text: 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 \(\Gamma_ n\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 4200466
zbMATH Keywords: spin group
zbMATH Keywords: symmetric group
zbMATH Keywords: vertex operators
zbMATH Keywords: symmetric algebra
zbMATH Keywords: Heisenberg algebra
zbMATH Keywords: double covering of the symmetric group
zbMATH Keywords: Fock space
zbMATH Keywords: symmetric polynomials
zbMATH Keywords: isometric isomorphism
zbMATH Keywords: Schur Q-function
zbMATH Keywords: irreducible characters
zbMATH Keywords: representation theory
reviewed by: Peter D Jarvis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4200466