Abstracts for the Seminar
Discrete Geometry and Combinatorics
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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