Diagrammatics for Soergel categories

TL;DR

This paper introduces a diagrammatic presentation of the monoidal category of Soergel bimodules for the symmetric group, providing a visual and algebraic framework for understanding their structure.

Contribution

It develops a planar diagrammatic calculus for Soergel bimodules specifically for the symmetric group, enhancing the understanding of their categorification of the Hecke algebra.

Findings

Diagrammatic presentation of Soergel bimodules for symmetric group

Simplified visualization of bimodule relations

Facilitates computations in categorification context

Abstract

The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.