Odd two-variable Soergel bimodules and Rouquier complexes

TL;DR

This paper explores the odd analogue of Soergel bimodules, developing categorical and graphical tools for two variables, and analyzing their properties and limitations in the context of Rouquier complexes and Reidemeister relations.

Contribution

It introduces the odd category of Soergel bimodules, establishes biadjointness, develops graphical calculus, and analyzes Rouquier complexes in the odd setting for two and three variables.

Findings

Biadjointness of functors established in the odd case.

Graphical calculus developed for the 2-variable odd Soergel category.

Invertibility of odd Rouquier complexes proven in the homotopy category.

Abstract

We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develop graphical calculi in the 2-variable case for the odd Soergel category and the related singular Soergel 2-category. We describe the odd analogue of the Rouquier complexes and establish their invertibility in the homotopy category. For three variables, the absence of a direct sum decomposition of the tensor product of generating Soergel bimodules presents an obstacle for the Reidemeister III relation to hold in the homotopy category.