On super Plucker embedding and cluster algebras

Duration: 1 hour 9 mins

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Description:	Dr Ekaterina Shemyakova (University of Toledo) 19th November 2021 \| 16:00 - 17:00


Created:	2021-11-22 14:41
Collection:	Modelling Behaviour to Inform Policy for Pandemics
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Dr Ekaterina Shemyakova
Language:	eng (English)


Abstract:	There has been active work towards the 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 Plucker 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 Plucker embedding” for 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 Plucker map takes the Grassmann supermanifold G_{r\|s}(V) to a “weighted projective space” P_{1,-1}(\Lambda^{r\|s}(V)\oplus \Lambda^{s\|rs}(\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 Plucker coordinates and show that our map is an embedding. We obtain the super analog of the Plucker relations and consider applications to conjectural super cluster algebras. (Based on joint work with Th. Voronov.)

Abstract:

There has been active work towards the 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 Plucker 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 Plucker embedding” for 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 Plucker map takes the Grassmann supermanifold G_{r|s}(V) to a “weighted projective space” P_{1,-1}(\Lambda^{r|s}(V)\oplus \Lambda^{s|rs}(\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 Plucker coordinates and show that our map is an embedding. We obtain the super analog of the Plucker relations and consider applications to conjectural super cluster algebras.

(Based on joint work with Th. Voronov.)

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