University of Minnesota Combinatorics Seminar: 2021-2022

Date: 11/05/2021

Speaker: Jose Simental Rodriguez

Braids, Weaves and Positroids
To a positive braid $\beta$ on $n$ strands we associate an affine algebraic variety $X(\beta)$ via an explicit set of equations. Equivalently, this variety can be thought of as the moduli space of tuples of flags in $n$-dimensional space satisfying some transversality conditions. Special cases of these varieties include open positroid and, more generally, open Richardson varieties. I will give some general properties of these varieties, like smoothness and dimension. Then I will construct algebraic open tori in them using a combinatorial object we call algebraic weaves. Conjecturally, these are cluster tori in a cluster structure on $X(\beta)$, and I will give examples where we can verify this conjecture. This is joint work with Roger Casals, Eugene Gorsky and Mikhail Gorsky.

Braids, Weaves and Positroids

To a positive braid $\beta$ on $n$ strands we associate an affine algebraic variety $X(\beta)$ via an explicit set of equations. Equivalently, this variety can be thought of as the moduli space of tuples of flags in $n$-dimensional space satisfying some transversality conditions. Special cases of these varieties include open positroid and, more generally, open Richardson varieties. I will give some general properties of these varieties, like smoothness and dimension. Then I will construct algebraic open tori in them using a combinatorial object we call algebraic weaves. Conjecturally, these are cluster tori in a cluster structure on $X(\beta)$, and I will give examples where we can verify this conjecture. This is joint work with Roger Casals, Eugene Gorsky and Mikhail Gorsky.