Strange Expectations in Affine Weyl Groups

TL;DR

This paper generalizes a combinatorial conjecture about the expected size of simultaneous cores to all affine Weyl groups, extending prior results and providing new models for coroot lattices.

Contribution

It extends Armstrong's conjecture proof to all affine Weyl groups and introduces combinatorial models for coroot lattices in classical and G2 types.

Findings

Generalization of the expected size of simultaneous cores to all affine Weyl groups

Refined notion of 'size' for simultaneous cores

New combinatorial models for coroot lattices in classical and G2 types

Abstract

Our main result is a generalization, to all affine Weyl groups, of P. Johnson's proof of D. Armstrong's conjecture for the expected number of boxes in a simultaneous core. This extends earlier results by the second and third authors in simply-laced type. We do this by modifying and refining the appropriate notion of the "size" of a simultaneous core. In addition, we provide combinatorial core-like models for the coroot lattices in classical type and type $G_2$.