Atomic length on Weyl groups

TL;DR

This paper introduces the atomic length statistic on Weyl groups, exploring its combinatorial and representation-theoretic properties, and connects it to classical enumeration problems and crystal theory.

Contribution

It defines the atomic length on Weyl groups and studies its properties, revealing interval structures and links to core partitions and crystal bases.

Findings

Atomic length exhibits properties similar to the usual length in finite Weyl groups.

In finite types, atomic length generally forms an interval, except in rank two.

In affine types, atomic length relates to classical enumeration problems like core partitions.

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.