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Twisted vertex representations via spin groups and the McKay correspondence. (English) Zbl 1100.17502

Summary: We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group \(\Gamma\) and a virtual character of \(\Gamma\), we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products \(\Gamma\wr \tilde {S}_n\) of \(\Gamma\) and a double cover of the symmetric group \(S_n\) for all \(n\). When \(\Gamma\) is a subgroup of \(\text{SL}_2(\mathbb {C})\) with the McKay virtual character, our construction gives a group-theoretic realization of the basic representations of the twisted affine and twisted toroidal Lie algebras. When \(\Gamma\) is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for \(\Gamma\wr \tilde {S}_n\).

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
05E05 Symmetric functions and generalizations
20C25 Projective representations and multipliers
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