Thick Soergel calculus in type A

TL;DR

This paper extends the diagrammatic presentation of Bott-Samelson bimodules to generalized Bott-Samelson bimodules in type A, providing a categorification of induced Hecke algebra representations using explicit idempotent descriptions.

Contribution

It introduces a diagrammatic categorification of generalized Bott-Samelson bimodules and their associated Hecke algebra representations, expanding previous work on Bott-Samelson bimodules.

Findings

Explicit description of idempotents for summands in bimodules

Diagrammatic categorification of induced Hecke algebra representations

Analysis of the reduced expression graph and higher Bruhat order

Abstract

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.