CANCELLED - Skein-triangulated representations of generalised braids

Abstract: Ordinary braid group $\mathrm{Br}_n$ is a well-known algebraic structure which encodes configurations of n non-touching strands ("braids") up to continuous transformations ("isotopies"). A classical result of Khovanov and Thomas states that there is a natural categorical action of $\mathrm{Br}_n$ on the derived category of the cotangent bundle of the variety of complete flags in $\mathbb{C}^n$. In this talk, I will introduce a new structure: the category $\mathrm{GBr}_n$ of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of $\mathrm{GBr}_n$. A decade-old conjecture states that there a skein-triangulated action of $\mathrm{GBr}_n$ on the cotangent bundles of the varieties of full and partial flags in $\mathbb{C}^n$. We prove this conjecture for $n = 3$. We also show that any categorical action of $\mathrm{Br}_n$ can be lifted to a skein-triangulated action of $\mathrm{GBr}_n$, which behaves like a categorical nil Hecke algebra. This is a joint work with Rina Anno and Lorenzo De Biase.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html