UW Combinatorics and Geometry Seminar: Soyeon Kim

Abstract: This talk focuses on Richardson varieties in the complete flag variety, which are intersections of a Schubert variety with an opposite Schubert variety. Richardson varieties are indexed by intervals in the Bruhat order on the symmetric group. In joint work with E. Gorsky and M. Sherman-Bennett, we study the following question: when is a Richardson variety a toric variety? All Richardson varieties admit an action by an (n-1)-dimensional torus in $SL_n$, called the standard torus, which also acts on the complete flag variety. The Richardson varieties which are toric varieties with respect to the standard torus were classified independently by Anderson and Tsukerman–Williams. We found that there are many additional toric Richardson varieties, which we call "unexpected”. In this rather expository talk, I will classify toric Richardson varieties using the combinatorics of Bruhat intervals and describing this "unexpected" torus action using 3D plabic graphs. If time permits, I will mention some recent developments and open questions in this direction.