On Tuesday, February 5 (my birthday!), I gave a talk titled “A diagrammatic representation of the Temperley-Lieb algebra” in the NAU Department of Mathematics and Statistics Colloquium. Here is the abstract.

One aspect of my research involves trying to prove that certain associative algebras can be faithfully represented using “diagrams.” These diagrammatic representations are not only nice to look at, but they also help us recognize things about the original algebra that we may not otherwise have noticed. In this talk, we will introduce the diagram calculus for the Temperley-Lieb algebra of type $A$. This algebra, invented by Temperley and Lieb in 1971, is a certain finite dimensional associative algebra that arose in the context of statistical mechanics in physics. We will show that this algebra has dimension equal to the nth Catalan number and discuss its relationship to the symmetric group. If time permits, we will also briefly discuss the diagrammatic representation of the Temperley-Lieb algebra of type affine $C$.

And here are the slides.

Despite the fact that this was a 50-minute talk, it was intended to be an overview of one aspect of a long and complex story. The subject matter is intimately related to my PhD thesis, as well as a series of papers that I have written.

- Ernst, D. C. (2010). Non-cancellable elements in type affine $C$ Coxeter groups.
*Int. Electron. J. Algebra*, 8, 191–218. [arXiv] - Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley-Lieb algebra, I.
*J. Pure Appl. Alg.*(to appear). [arXiv] - Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II. [arXiv]

In addition, there is (at least) a part III that goes with the last two papers that is in progress.

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

Website

928.523.6852

Twitter

Instagram

GitHub

arXiv

ResearchGate

Academia.edu

Mendeley

Google Scholar

Impact Story

ORCID

MAT 220: Math Reasoning

MAT 411: Abstract Algebra

This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

The source code is on GitHub.