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Book Chapter

How to draw Soergel bimodules


Patimo,  Leonardo
Max Planck Institute for Mathematics, Max Planck Society;

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Patimo, L. (2020). How to draw Soergel bimodules. In Introduction to Soergel bimodules (pp. 239-256). Cham: Springer.

Cite as: http://hdl.handle.net/21.11116/0000-0007-848F-3
In this chapter, we use the diagrammatic descriptions of morphisms between Bott–Samelson bimodules to give diagrammatic descriptions of two different bases for these bimodules. First, we describe the 01-basis, which is constructed by taking an iterated tensor product of a natural basis of Bs. The 01-basis is depicted diagrammatically as a sequence of straight vertical broken or unbroken lines. We define a commutative multiplication structure on Bott–Samelson bimodules, relate this multiplication to the diagrammatics of the 01-basis, and use these tools to define and prove the nondegeneracy of the global intersection form, a certain invariant R-bilinear form on Bott–Samelson bimodules. We then describe and give diagrammatics for another basis using light leaves. We show that this latter basis is compatible with the standard filtration on Bott–Samelson bimodules in a natural way, and we give a concrete example in the case of BS(s, t , s) with mst = 3.