Ted Voronov (Manchester)
The Berezinian of matrices, exterior powers and recurrence relations
The Berezinian is the analog of the determinant for the super case. Unlike determinant, it is a rational function of the matrix (not a polynomial). At the first glance, there is no connection of it with the exterior powers. In the supercase there are infinitely many exterior powers and no `top' power, in contrast with the familiar purely even situation. However, as we show, this connection exists and, loosely, reminds an analytic continuation from a neighborhood of zero to infinity.
We obtained universal recurrence relations that hold for supertraces of exterior powers of a linear operator; they are underlied by relations in Grothendieck ring. They come from expansions of the characteristic (rational) function Ber (1+z A) and lead to a new beautiful formula expressing the Berezinian as a ratio of Hankel determinants made of supertraces.
(Joint work with Hovhannes Khudaverdian.)