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Berezinians, Exterior Powers and Recurrent Sequences

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Abstract

We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V. We show that traces of exterior powers of A satisfy universal recurrence relations of period q. ‘Underlying’ recurrence relations hold in the Grothendieck ring of representations of GL(V). They are expressed by vanishing of certain Hankel determinants of order q+1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to express the Berezinian of an operator as a ratio of two polynomial invariants. We analyze the Cayley–Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer’s rule

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Authors and Affiliations

  1. School of Mathematics, University of Manchester, Sackville Street, Manchester, M60 1QD, UK

    H. M. Khudaverdian & TH. TH. Voronov

  2. Department of Theoretical Physics, Yerevan State University, 1 A. Manoukian Street, 375049, Yerevan, Armenia

    H. M. Khudaverdian

Authors
  1. H. M. Khudaverdian
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  2. TH. TH. Voronov
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Correspondence to TH. TH. Voronov.

Additional information

To the memory of Felix Alexandrovich Berezin

Mathematics Subject Classification (2000): 15A15, 58A50, 81R99

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Khudaverdian, H.M., Voronov, T.T. Berezinians, Exterior Powers and Recurrent Sequences. Lett Math Phys 74, 201–228 (2005). https://doi.org/10.1007/s11005-005-0025-7

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