## University of Sydney Algebra Seminar

# Oded Yacobi (University of Sydney)

## Friday 22 September, 12-1pm, Place: Carslaw 375

## 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