### Archives For Coxeter groups

On March 1-3, Jeff Rushall (he’s the Patrick Stewart-looking guy in the photo above) and I, together with a group of 8 Northern Arizona University mathematics majors, attended the 2013 Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at the University of New Mexico in Albuquerque. SUnMaRC is an annual conference that provides undergraduates in the Southwest an opportunity to give 20 minute talks in order to showcase the work they have been doing for their undergraduate research projects. Three of the 8 NAU students are currently conducting research projects under my guidance. That’s them in the picture above (taken during a break at the conference).

Two of my students, Dane Jacobson and Michael Woodward, have spent the spring semester studying the mathematics behind Spinpossible, which is a free game that is available for iOS and Android devices. Alternatively, you can just play the game in any modern web browser. The game is played on a 3 by 3 board of scrambled tiles numbered 1 to 9, each of which may be right-side-up or up-side-down. The objective of the game is to return the board to the standard configuration where tiles are arranged in numerical order and right-side-up. This is accomplished by a sequence of “spins”, each of which rotates a rectangular region of the board by 180 degrees. The goal is to minimize the number of spins used. It turns out that the group generated by the set of allowable spins is identical to the symmetry group of the 9 dimensional hyper-cube (equivalently, a Coxeter group of type $B_9$).

In a 2011 paper, Alex Sutherland and Andrew Sutherland (a father and son team) present a number of interesting results about Spinpossible and list a few open problems. You can find the paper here. As a side note, Alex is one of the developers of the game and his father, Andrew, is a mathematics professor at MIT. Using brute-force, the Sutherlands verified that every scrambled board can be solved in at most 9 moves. One of the goals of my students’ research project is to find a short proof of this fact. During their talk at SUnMaRC, Dane and Michael presented their current progress on this unexpectedly difficult problem. Here are their slides.

A week after SUnMaRC, Dane and Michael gave a similar and improved talk (which includes some random images of Samuel L. Jackson) during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for the FAMUS talks are undergraduates. You can find the slides for their FAMUS talk here.

My third student, Selina Gilbertson, has spent the spring semester working on extending the results of some of my previous students. During the 2011-2012 academic year, I mentored Ryan Cross, Katie Hills-Kimball, and Christie Quaranta at Plymouth State University on an original research project aimed at exploring the “T-avoiding” elements in Coxeter groups of type $F$. Coxeter groups can be thought of as generalized reflections groups. In particular, a Coxeter group is generated by a set of elements of order two. Every element of a Coxeter group can be written as an expression in the generators, and if the number of generators in an expression is minimal, we say that the expression is reduced. We say that an element $w$ of a Coxeter group is T-avoiding if $w$ does not have a reduced expression beginning or ending with a pair of non-commuting generators. My PSU students successfully classified the T-avoiding elements in the infinite Coxeter group of type $F_5$, as well as the finite Coxeter group of type $F_4$. At the time, we had conjectured that our classification holds more generally for arbitrary $F_n$. However, Selina has recently discovered that this is far from true. In Selina’s SUnMaRC talk, she relayed her current progress on classifying the T-avoiding elements in type $F_n$. Here are her slides.

Both sets of students gave wonderful talks. In fact, all 8 NAU students gave fantastic talks. I was impressed. Below are some photos from our trip.

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.

On Friday, September 12, 2012, I gave a 25 minute talk titled “An open problem of the symmetric group” during NAU’s Friday Afternoon Mathematics Undergraduate Seminar (FAMUS). Here is the open problem that I discussed.

How many commutation classes does the longest element in the symmetric group have?

The main goal of the talk was to understand what this question is asking. The secondary goal was to illustrate that mathematics is a lively field with open questions and to provide an example of what research in mathematics looks like. Here’s the abstract.

Many people are often surprised to hear that mathematicians do research. What is mathematical research? Research in mathematics takes many forms, but one common theme is that the research seeks to answer an open question concerning some collection of mathematical objects. The goal of this talk will be to introduce you to one of the many open questions in mathematics: how many commutation classes does the longest element in the symmetric group have? This problem has been nicknamed “Heroin Hero” by my advisor in honor of a game from the TV show “South Park” in which the character Stan obsesses over chasing a dragon. We will review the basics of the symmetric group and introduce all of the necessary terminology, so that we can understand this problem.

Here are the slides.

I really enjoy this short video introduction to group theory by James Grime (aka Singing Banana), especially since it mentions Coxeter groups (one of research interestes).