Soergel Calculus

TL;DR

This paper introduces a diagrammatic presentation of the monoidal category of Soergel bimodules, providing new proofs and bases for morphism spaces, which are fundamental in representation theory.

Contribution

It presents a generators-and-relations framework using planar diagrams for Soergel bimodules and offers a new proof of their classification.

Findings

Libedinsky's light leaves form a basis for morphism spaces

A new proof of Soergel's classification of indecomposable bimodules

Diagrammatic presentation simplifies understanding of the category

Abstract

The monoidal category of Soergel bimodules is an incarnation of the Hecke category, a fundamental object in representation theory. We present this category by generators and relations, using the language of planar diagrammatics. We show that Libedinsky's light leaves give a basis for morphism spaces and give a new proof of Soergel's classification of the indecomposable Soergel bimodules.