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SUMMARY:On super Plucker embedding and cluster algebras - Ekaterina Shemya
 kova (University of Toledo)
DTSTART:20211119T160000Z
DTEND:20211119T170000Z
UID:TALK164830@talks.cam.ac.uk
DESCRIPTION:There has been active work towards the definition of super clu
 ster algebras(Ovsienko\, Ovsienko-Shapiro\, and Li-Mixco-Ransingh-Srivasta
 va)\, but the notion isstill a mystery. As it is known\, the classical Plu
 cker map of a Grassmann manifold intoprojective space provides one of the 
 model examples for cluster algebras.In the talk\, we present our construct
 ion of &ldquo\;super Plucker embedding&rdquo\; for Grassmannian of r|s-pla
 nes in n|m-space. There are two cases. The first one is of completely even
  planes in a super space\, i.e. the Grassmannian G_{r|0}(n|m). It admits a
  straightforward algebraic construction similar to the classical case. In 
 the second\, general case of r|s-planes\, a more complicated construction 
 is needed. Our super Plucker map takes the Grassmann supermanifold G_{r|s}
 (V) to a &ldquo\;weighted projective space&rdquo\;&nbsp\; P_{1\,-1}(\\Lamb
 da^{r|s}(V)\\oplus \\Lambda^{s|rs}(\\Pi V))\, with weights +1\, &minus\;1.
  Here \\Lambda^{r|s}(V) denotes the (r|s)-th exterior power of a superspac
 e V and \\Pi is the parity reversion functor. We identify the super analog
  of Plucker coordinates and show that our map is an embedding. We obtain t
 he super analog of the Plucker relations and consider applications to conj
 ectural super cluster algebras.(Based on joint work with Th. Voronov.)
LOCATION:Seminar Room 2\, Newton Institute
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