Diagramatics of $A_1 \times A_1$ Folded Soergel Bimodules

TL;DR

This paper presents a local generators and relations presentation for the equivariantization of Soergel Bimodules of type A1×A1 under a Z/2 action, advancing the understanding of equivariant categories in this setting.

Contribution

It provides the first isotopy presentation of the equivariantized category of Soergel Bimodules of type A1×A1 using local generators and relations.

Findings

First isotopy presentation of the equivariant category

Explicit local generators and relations provided

Lays groundwork for studying equivariant Soergel Bimodules

Abstract

There is an action of $\mathbb{Z}/2$ on the category of Soergel Bimodules of type $A_1 \times A_1$ induced by the nontrivial automorphism of its Dynkin diagram. We give an isotopy presentation by local generators and relations of the equivariantization of the category of Soergel Bimodules of type $A_1 \times A_1$ under this action. This is the first step in a program to describe and study the categories of equivariant Soergel Bimodules.