# Oded Yacobi (University of Sydney)

## On equations defining the affine Grassmannian of $$SL_n$$

The affine Grassmannian $$Gr$$ of a semisimple group $$G$$ is an important infinite dimensional variety that appears in representation theory. This talk concerns the projective geometry of the affine Grassmannian when $$G=SL_n$$. More precisely, in this case $$Gr$$ naturally embeds into the Sato Grassmannian $$SGr$$, which is a limit of finite dimensional Grassmannians $$Gr(n,2n)$$ as $$n \to \infty$$, and we are interested in the equations defining the embedding $$Gr \subset SGr$$.

Kreiman, Lakshmibai, Magyar and Weyman constructed linear equations on $$SGr$$ which vanish on $$Gr$$ and conjectured that these equations suffice to cut out the affine Grassmannian. We recently proved this conjecture by reducing it to a question about finite dimensional Grassmannians. I’ll describe our method of proof and time permitting I’ll mention some conjectures (in both the finite and infinite dimensional settings) that arise from our work, and also some potential applications.

This is joint work with Dinakar Muthiah and Alex Weekes