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Orbifolds of symplectic fermion algebras


Authors: Thomas Creutzig and Andrew R. Linshaw
Journal: Trans. Amer. Math. Soc. 369 (2017), 467-494
MSC (2010): Primary 13A50, 17B69; Secondary 11F22, 17B65
DOI: https://doi.org/10.1090/tran6664
Published electronically: May 6, 2016
MathSciNet review: 3557781
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Abstract: We present a systematic study of the orbifolds of the rank $ n$ symplectic fermion algebra $ \mathcal {A}(n)$, which has full automorphism group $ Sp(2n)$. First, we show that $ \mathcal {A}(n)^{Sp(2n)}$ and $ \mathcal {A}(n)^{GL(n)}$ are $ \mathcal {W}$-algebras of type
$ \mathcal {W}(2,4,\dots , 2n)$ and $ \mathcal {W}(2,3,\dots , 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $ \mathcal {A}(mn)^{Sp(2n)}$ and $ \mathcal {A}(mn)^{GL(n)}$ for all $ m,n\geq 1$. We compute the characters of the irreducible representations of $ \mathcal {A}(mn)^{Sp(2n)\times SO(m)}$ and $ \mathcal {A}(mn)^{GL(n)\times GL(m)}$ appearing inside $ \mathcal {A}(mn)$, and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for $ \mathcal {A}(n)$; we show that for any reductive group $ G$ of automorphisms, $ \mathcal {A}(n)^G$ is strongly finitely generated


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Additional Information

Thomas Creutzig
Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: creutzig@ualberta.ca

Andrew R. Linshaw
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: andrew.linshaw@du.edu

DOI: https://doi.org/10.1090/tran6664
Received by editor(s): May 21, 2014
Received by editor(s) in revised form: January 8, 2015
Published electronically: May 6, 2016
Article copyright: © Copyright 2016 American Mathematical Society