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.