Theoretical Statistics and Mathematics Unit, ISI Delhi

August 12, 2020 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Sarjick Bakshi,
CMI Chennai

Title:
Torus quotients of Richardson varieties in the Grassmannian.

Abstract of Talk

We start recalling basics of Standard Monomial Theory for the Grassmannian variety and an introduction to Geometric Invariant theory (GIT). The GIT quotients of the Grassmannian variety and its subvarieties lead to many interesting geometric problems. One of the most important examples is to study the space of $n$ ordered points on the projective line modulo the automorphisms of the line. Howard, Milson, Snowden and Vakil showed that the ideal of relations is generated in degree at most four, and gave an explicit description of the generators of the ring of invariants for $n$ even and $r=2$ using graph theoretic methods. I will describe an alternative way to look at this problem for all $n$. We study the GIT quotient of the minimal Schubert variety in the Grassmannian admitting semistable points for the action of maximal torus $T$, with respect to the $T$-linearized line bundle ${\cal L}(n \omega_r)$ and show that this is smooth when $gcd(r,n)=1$. When $n=7$ and $r=3$ we study the GIT quotients of all Richardson varieties in the minimal Schubert variety. This talk is based on a joint work with Kannan and Subrahmanyam.