If W is a group generated by a subset T of its elements, the T-length 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: 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.