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The 2-Adic Representations of Klein’s Four Group

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Part of the Lecture Notes in Mathematics book series (LNM,volume 372)

Abstract

The indecomposable integral representations of Klein’s Four Group over a discrete valuation ring in which 2 is a non-zero prime element were first determined by Nazarova, as a consequence of her study [4] of modules over tetrad rings. It was shown in [2] that her results may be deduced from a functor-induced correspondence of representations with 4-subspace diagrams; this gives a little extra information about endomorphism rings, isomorphisms and direct decompositions. However, the correspondence used in [2] is rather awkward for computational purposes. This note contains a description of very much simpler and more explicit constructions for relating representations and diagrams. These are described in §1, which also contains formal statements of their properties. Their proofs occupy §2. As a result of recent work by Gelfand and Ponomarev [3] (working over an algebraically closed field) and Sheila Brenner [1] (over an arbitrary field), the structure of 4-subspace diagrams may be said to be completely understood. §3 contains a list of parameters (derived from [1]) sufficient to characterise all the indecomposable diagrams; normal forms for each of these indecomposables are given in [1].

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References

  1. Sheila Brenner, “On four subspaces of a vector space”, J. Algebra (to appear).

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  2. M.C.R. Butler, “Relations between diagrams of modules”, J. Lcndon Math. Soc. (2) 3 (1971), 577-587.Zbl.214,57.

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  3. I.M. Gelfand and V.A. Ponomarev, “Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space”, Hilbert space operators and operator algebras, pp. 163–237 (Colloquia Mathematica Societatis Janos Bolyai, 5. Tihany, Hungary, 1970; North-Holland, Amsterdam, London, 1972). Zb1. 238. 00011.

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  4. P.A. Hasaposa [L.A. Nazaroval, “ПредставЫения тетрадах” [Representations of tetrads], Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1361–1.378; Math. USSR Izv. 1 (1967), 1305–1321 (1969). MR36#6400.

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© 1974 Springer-Verlag Berlin Heidelberg

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Butler, M.C.R. (1974). The 2-Adic Representations of Klein’s Four Group. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_16

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  • DOI: https://doi.org/10.1007/978-3-662-21571-5_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06845-7

  • Online ISBN: 978-3-662-21571-5

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