Berezinians, Exterior Powers and Recurrent Sequences

Details Berezinians, Exterior Powers and Recurrent Sequences Berezinians, Exterior Powers and Recurrent Sequences

2003-09-10

arxiv.org/abs/math/0309188v2

Lett.Math.Phys. 74 (2005) 201-228

Mathematics

Differential Geometry

35 pages. LaTeX 2e. New version: paper substantially reworked and expanded, new results included

Scientific paper

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 explicitly express the Berezinian of an operator as a rational function of traces. 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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