Ekaterina Shemyakova (University of Toledo)
ekaterina.shemyakova@utoledo.edu
There has been active work towards definition of super cluster algebras (Ovsienko, Ovsienko-Shapiro, and Li-Mixco-Ransingh-Srivastava), but the notion is still a mystery. As it is known, the classical Pluecker map of a Grassmann manifold into projective space provides one of the model examples for cluster algebras.
We present our construction of a super Pluecker embedding for the Grassmannian of $r|s$-planes 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 Pluecker map takes the Grassmann supermanifold $G_{r|s}(V)$ to a weighted projective space $P_{1,-1}(\Lambda^{r|s}(V)\oplus \Lambda^{s|r}(\Pi V))$, with weights $+1$, $-1$. Here $\Lambda^{r|s}(V)$ denotes the $(r|s)$th exterior power of a superspace $V$ and $\Pi$ is the parity reversion functor. We identify the super analog of Pluecker coordinates and show that our map is an embedding. We obtain the super analog of the Pluecker relations and consider applications to conjectural super cluster algebras.
(Based on joint work with Th. Voronov.)