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Berezinians, exterior powers and recurrent sequences. (English) Zbl 1083.58008

Summary: 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

MSC:

58A50 Supermanifolds and graded manifolds
15A15 Determinants, permanents, traces, other special matrix functions
81R99 Groups and algebras in quantum theory
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