Diagrammatics for Soergel Categories

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.

Soergel calculus

“…The Frobenius biadjunction of B s with itself arises from certain cups and caps, which are constructed from trivalent vertices and dots. The one color isotopy relations (not drawn here; see [EK,(3.5) through (3.9)]), which guarantee that any diagram involving one color is isotopy invariant, follow from the axioms for Frobenius objects.…”

“…There it is explained that morphisms between Bott-Samelson bimodules in rank 2 are essentially governed by the TemperleyLieb algebra at a root of unity, a fact related to the (quantum) geometric Satake equivalence for sl 2 . The first important two color relation (not drawn here; see [EK,(3.27)] for an example) is the cyclicity of the 2m st -valent vertex, which is the last of the isotopy relations. (There is a subtlety if the Cartan matrix is not symmetric, which we ignore in the introduction.)…”

The two-color Soergel calculus

“…For this reason, this paper is not the easiest introduction to Soergel theory in general, or to the diagrammatic style of the results we present. The novice should perhaps begin by reading about Soergel bimodules in type A in [9].…”

Schubert calculus and torsion explosion

“…Recently Elias and the author found a presentation for the monoidal category of Soergel bimodules by generators and relations [EW], building on the work of Libedinsky [Lib10], Elias-Khovanov [EK10] and Elias [Eli16]. One of the applications of this theory is that one can decide whether a given intersection cohomology complex has p-torsion in its stalks or costalks (the bridge between intersection cohomology and Soergel bimodules is provided by the theory of parity sheaves).…”