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The two-color Soergel calculus

Part of: Representation theory of groups General commutative ring theory

Published online by Cambridge University Press:  22 September 2015

Ben Elias
Ben Elias*
Affiliation:
Math Department, Fenton Hall, University of Oregon, Eugene, OR 97403, USA email belias@uoregon.edu
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Abstract

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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 $\mathfrak{sl}_{2}$. 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 $W$.

Keywords

dihedral groupHecke algebraSoergel bimodulesTemperley–Lieb algebraJones–Wenzl projectorgeometric Satakediagrammatics

MSC classification

Primary: 20C08: Hecke algebras and their representations
Secondary: 13A50: Actions of groups on commutative rings; invariant theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 2 , February 2016 , pp. 327 - 398
Copyright
© The Author 2015 

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