- Title: Cellular structures using -tilting modules
- Authors: Henning Haahr Andersen, Catharina Stroppel and Daniel Tubbenhauer
- Status: Pacific J. Math. 292-1 (2018), 21-59. 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
We use the theory of -tilting modules
to construct cellular bases for
algebras. Our methods are quite general and work for any quantum group
to a Cartan matrix
and include the non-semisimple 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.
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.
- The group algebras of the symmetric group
and its corresponding Iwahori-Hecke algebra .
- related algebras
like Temperley-Lieb algebras and others.
- Spider algebras in the sense of Kuperberg.
- The group algebras of the complex reflection groups
and its corresponding Ariki-Koike 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
- Generalized Khovanov arc algebras.
- -web algebras.
- Cyclotomic Khovanov-Lauda Rouquier algebras of type .
Our construction of cellular bases is explicit and can be illustrated in a “bow-tie” diagram.
Moreover, for the Temperley-Lieb we obtain the so-called “generalized Jones-Wenzl projectors” as
basis elements, e.g. (up to a scalar) such a projector looks like: