9 research outputs found
A Diagrammatic Temperley-Lieb Categorification
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 cell
modules of the Temperley-Lieb algebra.Comment: long awaited update to published versio
Thick Soergel calculus in type A
Let R be the polynomial ring in n variables, acted on by the symmetric group
S_n. 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 S_n, and the semi-orientation on this graph
given by the higher Bruhat order of Manin and Schechtman.Comment: Changed title. Expanded the exposition of the main proof. This paper
relies extensively on color figure