270 research outputs found
Diagrammatics for Coxeter groups and their braid groups
We give a monoidal presentation of Coxeter and braid 2-groups, in terms of
decorated planar graphs. This presentation extends the Coxeter presentation. We
deduce a simple criterion for a Coxeter group or braid group to act on a
category.Comment: Many figures, best viewed in color. Minor updates. This version
agrees with the 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
Diagrammatics and the Proactive Visualization of Legal Information
This article performs an analysis of one mode of visual legal communication:
diagrammatics and the visualization of legal data and other information
in legal instruments and communications.
The Proactive Law movement and the Legal Design movement each
seek to transform legal instruments and documents to improve access to and
comprehension of the communication of law to all persons. “All persons”
includes both law-trained and non-law-trained persons and extends from the
literate and educated all the way to disadvantaged, illiterate, and less-thanfully
literate persons. The overall goal of the Proactive Law movement and a
primary goal of Legal Design is to improve the understanding of legal
rights, relationships, and obligations expressed in legal products, instruments,
services, processes, and systems through illustration, simplification,
engagement, and inclusiveness in the text and visual components of these
instruments and communications
Planar diagrammatics of self-adjoint functors and recognizable tree series
A pair of biadjoint functors between two categories produces a collection of
elements in the centers of these categories, one for each isotopy class of
nested circles in the plane. If the centers are equipped with a trace map into
the ground field, then one assigns an element of that field to a diagram of
nested circles. We focus on the self-adjoint functor case of this construction
and study the reverse problem of recovering such a functor and a category given
values associated to diagrams of nested circles.Comment: 61 pages. v2: This revised version is to appear in Pure Appl. Math.
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