How to draw Soergel bimodules

Patimo, Leonardo

doi:10.1007/978-3-030-48826-0_12

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How to draw Soergel bimodules

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

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https://doi.org/10.1007/978-3-030-48826-0_12
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Citation

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

Cite as: https://hdl.handle.net/21.11116/0000-0007-848F-3

Abstract

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.