1 research outputs found
On super Pl\"{u}cker embedding and cluster algebras
We define a super analog of the classical Pl\"{u}cker embedding of the
Grassmannian into a projective space. The difficulty of the problem is rooted
in the fact that super exterior powers are not a simple
generalization from the completely even case (this works only for when it
is possible to use ). To construct the embedding we need to
non-trivially combine a super vector space and its parity-reversion . Our "super Pl\"{u}cker map" takes the Grassmann supermanifold
to a "weighted projective space" with weights . A simpler map works for the case . We construct a super analog of
Pl\"{u}cker coordinates, prove that our map is an embedding, and obtain "super
Pl\"{u}cker relations". It is interesting that another type of relations (due
to Khudaverdian) is equivalent to the (super) Pl\"{u}cker relations in the case
. We discuss application to much sought-after super cluster algebras
and construct a super cluster structure for and
.Comment: LaTeX, 56 pp. Exposition reworked and new results include