Dual pairs on spinors. Cases of \((C_ m,C_ n)\) and \((C_ m^{(1)},C_ n^{(1)})\)
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.
spinor versions of Weyl's reciprocity theorem
scientific article; zbMATH DE number 4044049