Reflection length in affine reflection groups 9/6/2017

Speaker: Joel Brewster Lewis, GWU

Abstract: If $W$ is a group generated by a subset $T$ of its elements, the $T$ -length $\ell_T(w)$ of an element $w$ in $W$ is the smallest integer $k$ such that $w$ can be written as a product of $k$ elements of $T$ . In the case that $W$ is a finite Coxeter group (or the real orthogonal group, or the general linear group of a finite-dimensional vector space) with $T$ equal to the set of all reflections in $W$ , this extrinsic notion can be given an intrinsic, geometric definition: the reflection length of $w$ is equal to the codimension of the fixed space, or equivalently to the dimension of its moved space. In the case that $W$ is the symmetric group $S_n$ , reflection length also has a combinatorial interpretation: $\ell_T(w) = n - c(w)$ , where $c(w)$ is the number of cycles of $w$ .

In this talk, we'll describe a new result (joint with Jon McCammond, Kyle Petersen, and Petra Schwer) extending these results to the case of affine Coxeter groups. Our formula, which has a short, uniform proof, involves two different notions of dimension for the moved space of a group element. In the case that the group is the affine symmetric group, we also give a combinatorial interpretation for these dimensions.