Thick Soergel calculus in type A - Mathematics > Representation TheoryReport as inadecuate

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Abstract: Let R be the polynomial ring in n variables, acted on by the symmetric groupS n. Soergel constructed a full monoidal subcategory of R-bimodules whichcategorifies the Hecke algebra, whose objects are now known as Soergelbimodules. Soergel bimodules can be described as summands of Bott-Samelsonbimodules attached to sequences of simple reflections, or as summands ofgeneralized Bott-Samelson bimodules attached to sequences of parabolicsubgroups. A diagrammatic presentation of the category of Bott-Samelsonbimodules 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-Samelsonbimodules. We also diagrammatically categorify the representations of the Heckealgebra which are induced from trivial representations of parabolic subgroups.The main tool is an explicit description of the idempotent which picks out ageneralized Bott-Samelson bimodule as a summand inside a Bott-Samelsonbimodule. This description uses a detailed analysis of the reduced expressiongraph of the longest element of S n, and the semi-orientation on this graphgiven by the higher Bruhat order of Manin and Schechtman.

Author: Ben Elias


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