Abstracts
for the Seminar

Fall 2017

**Speaker: **Kyle Petersen, DePaul University

**Title: **Reflection length in Coxeter groups

**Time:** 2:30 PM, Monday, November 27, 2017

**Place:** Malott 206

**Abstract:** Every element in a Coxeter group can be expressed as a
product of reflections. The minimal number of reflections needed to
express the element $w$ is the *reflection length* of $w$. Reflection
length is additive with respect to direct products of groups, so it
suffices to consider reflection length in irreducible cases. Here we
have a trichotomy result about reflection length that ought to be
better known:
1) If $W$ is a spherical (finite) Coxeter group of rank $n$, reflection
length is bounded by $n$.
2) If $W$ is an affine Coxeter group of rank $n$, reflection length is
bounded by $2n$.
3) If $W$ is otherwise, reflection length is unbounded.
In the 1970s, Carter gave a geometric characterization of reflection
length in the spherical case; the length of an element $w$ is the
dimension of the "move-set" of $w$. I will describe a similar geometric
characterization of reflection length in the affine case, along with
some remarks about computing reflection length.
This is joint work with Joel Lewis, Jon McCammond, and Petra Schwer.

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