

Cellular structures using tilting modules
 Title: Cellular structures using tilting modules
 Authors: Henning Haahr Andersen, Catharina Stroppel and Daniel Tubbenhauer
 Status: Pacific J. Math. 2921 (2018), 2159. Last update: Mon, 2 Oct 2017 13:46:03 GMT
 ArXiv link: http://arxiv.org/abs/1503.00224
 ArXiv version = 0.99 published version
 LaTex Beamer presentation: Slides1, Slides2, Slides3
 Additional file: File
Abstract
We use the theory of tilting modules
to construct cellular bases for
centralizer
algebras. Our methods are quite general and work for any quantum group
attached
to a Cartan matrix
and include the nonsemisimple cases for being a root of unity and ground
fields of positive characteristic.
Our approach also
generalize to certain
categories containing infinite dimensional modules.
As applications, we give a new
semisimplicty criterion for centralizer
algebras, and recover the cellularity of several known algebras
(with partially new cellular bases) which all fit into our general setup.
A few extra words
One of our main points is that our approach collects cellular structures on a
lot of important algebras under one roof (usually the proofs
of cellularity of these algebras are spread over the literature). For example, we
recover the cellularity of the
following interesting algebras.
 The group algebras of the symmetric group
and its corresponding IwahoriHecke algebra .
 related algebras
like TemperleyLieb algebras and others.
 Spider algebras in the sense of Kuperberg.
 The group algebras of the complex reflection groups
and its corresponding ArikiKoike algebra .
In particular, Hecke algebras of type .
 Algebras related to , e.g. (quantum) rook monoid
algebras and blob algebras
.
 Brauer algebras and its quatazation, the
BMW algebras .
 Algebras related to , e.g. walled Brauer
algebras .
 More...
Even better: our methods also generalize to categories containing infinite dimensional modules.
In particular, to the BGG category ,
its parabolic subcategories and
its quantum cousin . Using this, we recover for
example the following cellular structures as well.
 Generalized Khovanov arc algebras.
 web algebras.
 Cyclotomic KhovanovLauda Rouquier algebras of type .
 algebras.
 More...
Our construction of cellular bases is explicit and can be illustrated in a “bowtie” diagram.
Moreover, for the TemperleyLieb we obtain the socalled “generalized JonesWenzl projectors” as
basis elements, e.g. (up to a scalar) such a projector looks like:

NEWS

I am still a fool.

My paper
got accepted.

The arXiv version of this paper
was updated.

My paper
got accepted.

The arXiv version of this paper
was updated.
"There are two ways to do mathematics.
The first is to be smarter than everybody else.
The second way is to be stupider than everybody else  but persistent." 
based on a quotation from Raoul Bott.
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