Dual pairs on spinors. Cases of \((C_ m,C_ n)\) and \((C_ m^{(1)},C_ n^{(1)})\)
type: Item
instance of: scholarly article
title: Dual pairs on spinors. Cases of \((C_ m,C_ n)\) and \((C_ m^{(1)},C_ n^{(1)})\)
zbMATH Open document ID: 0641.17009
author: Kohji Hasegawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: Inspired by the ``rank-level duality'' in the Goddard-Kent-Olive construction of the Virasoro algebra representations, one presents here spinor versions of Weyl's reciprocity theorem which originally says that the symmetric group \({\mathfrak S}_ m\) and the general linear group GL(V) are mutually commutant (form a dual pair in the sense of R. Howe) on the tensor module \(V^{\otimes m}.\) We will denote by \({\mathfrak g}^{\wedge}\) the affine Kac-Moody Lie algebra associated to a complex simple Lie algebra \({\mathfrak g}\). The main theorems of this paper are that \({\mathfrak sp}(V_ 1)\) and \({\mathfrak sp}(V_ 2)\) (resp. \({\mathfrak sp}(V_ 1)^{\wedge}\) and \({\mathfrak sp}(V_ 2)^{\wedge})\) are mutually commutant on the \({\mathfrak o}(V_ 1\otimes V_ 2)\)- (resp. \({\mathfrak o}(V_ 1\otimes V_ 2)^{\wedge}\)-) spinors (that is, the fermion Fock space). Moreover the highest weight vectors with respect to \({\mathfrak sp}(V_ 1)\oplus {\mathfrak sp}(V_ 2)\) (resp. \({\mathfrak sp}(V_ 1)^{\wedge}\oplus {\mathfrak sp}(V_ 2)^{\wedge})\) themselves are expressed by means of Young diagrams. Varying \(V_ 2\), we get any integrable highest weight \({\mathfrak sp}(V_ 1)\)- (resp. \({\mathfrak sp}(V_ 1)^{\wedge}\)-) module as an irreducible component of the spin modules. The notion of the seesaw pair (terminology owing to S. Kudla) is useful to understand dualities among branching rules. From this viewpoint, the results in this paper give an illustrated explanation and generalizations of the Jimbo-Miwa dualities on the branching rules [\textit{M. Jimbo} and \textit{T. Miwa}, Adv. Stud. Pure Math. 6, 17-65 (1985; Zbl 0579.17009)], equivalences among the coset representations of the Virasoro algebra. A detailed paper is submitted to Publ. RIMS, Kyoto Univ.. It contains similar results for any simple or affine Lie algebras of classical type.
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4044049
zbMATH Keywords: spinor versions of Weyl's reciprocity theorem
zbMATH Keywords: dual pair
zbMATH Keywords: affine Kac- Moody Lie algebra
zbMATH Keywords: commutant
zbMATH Keywords: fermion Fock space
zbMATH Keywords: highest weight vectors
zbMATH Keywords: Young diagrams
zbMATH Keywords: seesaw pair
zbMATH Keywords: dualities
zbMATH Keywords: branching rules
zbMATH Keywords: coset representations
zbMATH Keywords: Virasoro algebra
MaRDI profile type: Publication
full work available at URL: pjaa.63.400
OpenAlex ID: W2001349093
cites work: Spinor representations of affine Lie algebras
cites work: Q3223062
cites work: Q3701597
cites work: Q3693672
cites work: Modular and conformal invariance constraints in representation theory of affine algebras
DOI: 10.3792/PJAA.63.400
description: scientific article; zbMATH DE number 4044049