Abstract
The level (2, 2)-Heisenberg GroupG(2, 2) as first introduced by Mumford in [Mu] is a subgroup inSL(4,ℂ) of order 32. LetN be the normalizer ofG(2, 2) inSL(4,ℂ).
This note describes explicitely the two natural isomorphisms fromN/G(2, 2) to the symmetric group\(\mathbb{S}_6 \) of 6 elements. These identifications clarify the computations in the classical treatises as for example in the theory of Kummer surfaces in [Hu] and the theory of the quadric line complex as in [Je] and will be used in [Nie] to describe the moduli space for abelian surfaces with a level (2, 6)-structure.
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References
[Bo] Borel, A.: Linear Algebraic Groups: New York W.A. Benjamin 1969
[Di] Dickson, L.: Linear Groups: New York Dover Publications 1958
[H] Hudson, R. W. H. T.: Kummer's Quartic Surface: Cambridge University Press 1905
[J] Jessop, C. M.: A Treatise on the Line Complex: New York Chelsea 1903
[Li] Littlewood, D. E.: The Theory of Group Characters: Oxford Clarendon Press 1958
[Mu] Mumford, D.: On the Equations defining Abelian Varieties I. Invent. Math., 1 287–354 (1966)
[Nie] Nieto, I.: Abelian Surfaces with a level (2, 6)-structure. Forthcoming
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Nieto, I. The normalizer of the level (2,2)-Heisenberg Group. Manuscripta Math 76, 257–267 (1992). https://doi.org/10.1007/BF02567760
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DOI: https://doi.org/10.1007/BF02567760