Atomic length on Weyl groups

Abstract

We define a new statistic on Weyl groups called the atomic length and investigate its combinatorial and representation-theoretic properties. In finite types, we show a number of properties of the atomic length which are reminiscent of the properties of the usual length. Moreover, we prove that, with the exception of rank two, this statistic describes an interval. In affine types, our results shed some light on classical enumeration problems, such as the celebrated Granville–Ono theorem on the existence of core partitions, by relating the atomic length to the theory of crystals.