# Shemyakova E. On super Plücker embedding and cluster algebras (abstract)

Speaker: Ekaterina Shemyakova

Title: On super Plücker embedding and cluster algebras

Abstract:
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 Plücker map of a Grassmann manifold into projective space provides one of the model examples for cluster algebras. In the talk, we present our construction of "super Plücker embedding" for Grassmannian of ${\displaystyle r|s}$-planes in ${\displaystyle n|m}$-space.

There are two cases. The first one is of completely even planes in a super space, i.e., the Grassmannian ${\displaystyle G_{r|0}(n|m)}$. It admits a straightforward algebraic construction similar to the classical case. In the second, general case of ${\displaystyle r|s}$-planes, a more complicated construction is needed.

Our super Plücker map takes the Grassmann supermanifold ${\displaystyle G_{r|s}(V)}$ to a "weighted projective space" ${\displaystyle P_{1,-1}(\Lambda ^{r|s}(V)\oplus \Lambda ^{s|rs}(\Pi V))}$, with weights ${\displaystyle +1,-1}$. Here ${\displaystyle \Lambda ^{r|s}(V)}$ denotes the ${\displaystyle (r|s)}$th exterior power of a superspace ${\displaystyle V}$ and ${\displaystyle \Pi }$ is the parity reversion functor. We identify the super analog of Plücker coordinates and show that our map is an embedding. We obtain the super analog of the Plücker relations and consider applications to conjectural super cluster algebras.

Based on a joint work with Th. Voronov.

Video

Event: Diffieties, Cohomological Physics, and Other Animals, 13-17 December 2021, Moscow.
Alexandre Vinogradov Memorial Conference.