Title: Macdonald polynomials as a sum of affine Demazure characters
Abstract: It is known that in type $A_{n}^{(1)}$ and simply-laced types, the specialized Macdonald polynomials at $t=0$ is an affine Demazure character. In type $C_{n}^{(1)}$, there is no explicit formula describing this relationship. We begin by providing the necessary background and summarize known results in type $A_{n}^{(1)}$. We then motivate and propose conjectures for the relationship between the specialized Macdonald polynomial and Demazure characters in type $C_{n}^{(1)}$.
Title: Frozen variables for open Richardson varieties
Abstract: The open Richardson varieties $R_{v,w}^{\circ}$ are first introduced by Kazhdan and Lusztig in 1979. It is an intersection of Schubert cell $X_{w}^{\circ}$ and an opposite Schubert cell $(X^{v})^{\circ}$. By the work of Casals-Gorsky-Gorsky-Le-Shen-Simental and Galashin-Lam-Sherman-Bennett-Speyer, there is a cluster structure on open Richardson varieties. A cluster structure allows one to define mutable and frozen cluster variables which are certain functions on $R_{v,w}^{\circ}$. In this talk, I will discuss two problems related to the frozen variables for open Richardson varieties. In particular, we will see the combinatorial description for the number of frozens $f_{v,w}$ in a quiver of $R_{v,w}^{\circ}$ and that how a torus of dimension equal to $f_{v,w}$ acts on $R_{v,w}^{\circ}$.