Super Plücker embedding and super cluster algebras
There has been active work towards definition of super cluster algebras (Ovsienko, Ovsienko- Shapiro, Li-Mixco-Ransingh-Srivastava, and Musiker-Ovenhouse-Zhang), but the notion is still a mystery. In our approach we start with one of the key model examples for cluster algebras — classical Pluecker embedding. In the talk, I present our construction of “super Pluecker embedding” and corresponding “super Pluecker relations” for Grassmannian of r|s-planes in n|m-space. (Only a very special case was considered before in the literature, namely, of 2|0- planes in 4|1-space, by Cervantes-Fioresi-Lledo.) The straightforward algebraic construction of exterior powers goes through for the Grassmannian Gr|0(n|m), i.e. completely even planes in the superspace. For the general case of r|s-planes, a more complicated construction is needed. Our super Pluecker map takes the Grassmann supermanifold Gr|s(V ) to a “weighted projective space” P1,−1(Λr|s(V )⊕Λs|rs(ΠV )), with weights +1, 1. Here Λr|s(V ) denotes the (r|s)th exterior power of a superspace V and Π is the parity reversion functor. We identify the super analog of Pluecker coordinates and show that our map is an embedding. Based on the obtained super analogs of the Pluecker relations for the particular case of G2|0(n|1), we propose a super cluster structure which can be described by a simple mnemonic rule. (Based on a joint work with Th. Voronov.)