On super Plücker embedding and cluster algebras Article (Faculty180)

cited authors

  • Shemyakova, Ekaterina; Voronov, Theodore


  • We define a super analog of the classical Pl\"{u}cker embedding of the Grassmannian into a projective space. Only a very special case was considered before in the literature. The "super Pl\"{u}cker map" that we introduce takes the Grassmann supermanifold $G_{r|s}(V)$ to a "weighted projective space" $P\left(\Lambda^{r|s}(V)\oplus \Lambda^{s|r}(\Pi V)\right)$ with weights $+1,-1$. Here $\Lambda^{r|s}(V)$ denotes the $r|s$th exterior power of a superspace $V$. We identify the super analog of Pl\"{u}cker coordinates and show that our map is a rational embedding. We investigate a super analog of the Pl\"{u}cker relations. We obtain them for $r|0$ and $n|m$. Also, we consider another type of relations due to H. Khudaverdian and show that they are equivalent to (super) Pl\"{u}cker relations for $r|s=2|0$ (this is new even in the classical case), but in general are only a consequence of the Pl\"{u}cker relations. We also discuss possible application to super cluster algebras (the notion only partly known at present). [Journal_ref: ]

publication date

  • 2022

published in


  • 28