Potential Summer 2014 REU Project

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The combinatorial nature of my research naturally lends itself to collaborations with undergraduates, and my goal is to incorporate students in my research as much as possible. Below I outline a potential REU project for 1-4 students to work on during the summer of 2014. For other examples of research projects that I have conducted with undergraduates, see my undergraduate research page.

Project Overview

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In mathematics, one uses groups to study symmetry. In particular, a reflection group is used to study the reflection and rotational symmetry of an object. A Coxeter group can be thought of as a generalized reflection group, where the group is generated by a set of elements of order two (i.e., reflections) and there are rules for how the generators interact with each other.

More specifically, a Coxeter system $(W,S)$ consists of a group $W$ (called a Coxeter group) generated by a set $S$ of involutions with presentation
$$W = \langle S : s^{2}=1, (st)^{m(s, t)} = 1 \rangle,$$
where $m(s, t) \geq 2$ for $s\neq t$. Since $s$ and $t$ are involutions, the relation $(st)^{m(s,t)}=1$ can be rewritten as $sts \cdots=tst\cdots$, where each product contains $m(s,t)$ factors. These relations are referred to as braid relations and when $m(s,t)\geq 3$, the corresponding relation is called a long braid relation. Note that if $m(s,t)=2$, then $s$ and $t$ commute. An expression $s_{x_1}s_{x_2}\cdots s_{x_m}$ in the generators of $W$ is reduced if $m$ is minimal.

A classic result of Coxeter groups, known as Matsumoto’s theorem, states that any two reduced expressions of the same element differ by a sequence of braid relations. Conjugating an expression by an initial generator results in a cyclic shift of the corresponding word: $s_{x_1}(s_{x_1}s_{x_2}\cdots s_{x_m})s_{x_{1}}= s_{x_2}s_{x_3}\cdots s_{x_m}s_{x_1}$. An expression is called cyclically reduced if every cyclic shift of it is reduced, and we ask the following question.

Do two cyclically reduced expressions of conjugate elements differ by a sequence of braid relations and cyclic shifts?

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An affirmative answer would be a cyclic version of Matsumoto’s theorem (CVMT) and would provide an algorithmic solution to the conjugacy problem for Coxeter groups. While the answer to this question is, in general, “no,” it seems to “often be true,” and understanding when the answer is “yes” is a central focus of a broad ongoing research project together with Richard M.Green (University of Colorado, Boulder) and Matthew Macauley (Clemson University). In 2009, the Erikssons showed that the CVMT holds for Coxeter elements (i.e., those elements whose reduced expressions contain a unique occurrence of each generator from $S$) [2]. Key to this was establishing necessary and sufficient conditions for a Coxeter element $w\in W$ to be logarithmic; that is, for $\ell(w^k)=k\cdot \ell(w)$ to hold for all $k\geq 1$. Trying to understand which elements in a Coxeter group are logarithmic is one of the broad goals of the proposed research.

An element $w$ is fully commutative (FC) if any two of its reduced expressions are equivalent by iterated commutations. Equivalently, $w$ is FC if and only if every reduced expression “avoids long braid relations” [3]. An element $w$ is cyclically fully commutative (CFC) if every cyclic shift of every reduced expression for $w$ is a reduced expression for an FC element.

In a recent paper, we showed that the CFC elements with full support (i.e., every generator appears at least once in every reduced expression) in infinite Coxeter groups that are absent of a feature called a “large band” are logarithmic [1]. However, evidence suggests that the result holds more generally. The CFC elements are rich in combinatorics and will be the central focus of the proposed research. In particular, the research team will tackle some subset of the following problems.

  1. Identify a larger class of CFC elements that are logarithmic (i.e., loosen the “large band” restriction).
  2. Identify necessary and sufficient conditions for CFC elements in groups with finitely many CFC elements to be conjugate.
  3. Characterize and enumerate the conjugacy classes of CFC elements groups with finitely many CFC elements.

Prerequisites

Perhaps surprisingly, one advantage of the proposed research is that it does not require much prerequisite knowledge. While a background in abstract algebra would be helpful, it is not necessary. If the above problems prove to be too difficult or too easy, there are many more related problems for the students to work on.

References

[1] T. Boothby, J. Burkert, M. Eichwald, D.C. Ernst, R. M. Green, and M. Macauley. On the cyclically fully commutative elements of Coxeter groups. J. Algebraic Combin., 36(1), 2012. [PDF]

[2] H. Eriksson and K. Eriksson. Conjugacy of Coxeter elements. Elect. J. Comb., 16(2), 2009. [PDF]

[3] J. R. Stembridge. On the fully commutative elements of Coxeter groups. J. Algebraic Combin., 5(4), 1996.

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