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Let R be the polynomial ring in n variables, acted on by the symmetric group Sn. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we extend it to a presentation of the category of generalized Bott-Samelson bimodules. We also diagrammatically categorify the representations of the Hecke algebra which are induced from trivial representations of parabolic subgroups. The main tool is an explicit description of the idempotent which picks out a generalized Bott-Samelson bimodule as a summand inside a Bott-Samelson bimodule. This description uses a detailed analysis of the reduced expression graph of the longest element of Sn, and the semi-orientation on this graph given by the higher Bruhat order of Manin and Schechtman. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures. [ABSTRACT FROM AUTHOR]

We compute the trace decategorification of the Hecke category for an arbitrary Coxeter group. More generally, we introduce the notion of a strictly object-adapted cellular category and calculate the trace for such categories. [ABSTRACT FROM AUTHOR]

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 $W$. [ABSTRACT FROM AUTHOR]

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1- skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the irreduciblemodules of the Temperley-Lieb algebra. [ABSTRACT FROM AUTHOR]