The two-color Soergel calculus

Abstract

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W . The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is infinite, the parameter q of the Temperley-Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for sl(2). When W is finite, q must be specialized to an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soergel bimodule for the longest element of W .

Revised, this is essentially the published version

References in corpus (4)

• The Karoubi envelope and Lee's degeneration of Khovanov homology
• A Formula for the Jones-Wenzl Projections
• Ideals in the Temperley Lieb Category
• On cubes of Frobenius extensions

Cited by in corpus (30)

• Simple transitive 2-representations via (co)algebra 1-morphisms
• Schubert calculus and torsion explosion
• Categorification of quantum symmetric pairs I
• Trace decategorification of the Hecke category
• Modular representation theory in type A via Soergel bimodules
• Two-color Soergel calculus and simple transitive 2-representations
• Diagrams and discrete extensions for finitary 2-representations
• The Modular Temperley-Lieb Algebra
• Existence and rotatability of the two-colored Jones-Wenzl projector
• Soergel calculus and Schubert calculus
• $(\ell ,p)$-Jones-Wenzl Idempotents
• Blob algebra approach to modular representation theory
• Mixed modular perverse sheaves on moment graphs
• Monoidal categories, representation gap and cryptography
• The homology of the Temperley-Lieb algebras
• Koszul Duality and Soergel Bimodules for Dihedral Groups
• The 2-braid group and Garside normal form
• Asymptotics in finite monoidal categories
• Cellularity of the p-Canonical Basis for Symmetric Groups
• Soergel bimodules for universal Coxeter groups
• Blob algebra and two-color Soergel calculus
• Localized calculus for the Hecke category
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
• Classification problems in 2-representation theory
• Categorifying Hecke algebras at prime roots of unity, part I
• Extreme representations of semirings
• Jucys-Murphy elements for Soergel bimodules
• Twisted actions on cohomologies and bimodules
• Presenting Hecke endomorphism algebras by Hasse quivers with relations
• Ext-enhanced monoidal Koszul duality for ${\mathrm{GL}}_2$