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The two-color Soergel calculus
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 2 / February 2016
- Published online by Cambridge University Press:
- 22 September 2015, pp. 327-398
- Print publication:
- February 2016
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- Article
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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 
$W$.
