Abstract
Diagrammatic algebra provides a useful way to study Soergel bimodules. This approach proceeds via the simpler category [Formula: see text]Bim of Bott–Samelson bimodules, for which there is a well developed diagrammatic calculus. As Soergel bimodules are summands of Bott–Samelson bimodules, it is important to understand idempotents in the category [Formula: see text]Bim. For Coxeter groups of type [Formula: see text], we analyze this problem for certain important idempotents, namely, the idempotent projecting to the indecomposable Soergel bimodule corresponding to the longest element of the Coxeter group. We use a strategy analogous to one used by Elias in type [Formula: see text]. We present a full explicit calculation for the first nontrivial case [Formula: see text]. The strategy is applicable for general [Formula: see text] but much more involved. We hope that the [Formula: see text] case serves as a stepping stone for the general case.
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