Looping of the numbers game and the alcoved hypercube

TL;DR

This paper analyzes the looping case of Mozes's numbers game related to affine Weyl groups, revealing a new poset structure, a partition of the Weyl group, and connections to hypercube triangulations, with explicit polynomial computations.

Contribution

It introduces a new partition of the Weyl group into graded posets linked to extended Dynkin diagrams and establishes their properties and dualities.

Findings

All configurations in the orbit are reachable via the numbers game.

The posets are selfdual and isomorphic, reflecting hypercube triangulations.

The top degree of the posets is cubic in the number of vertices.

Abstract

We study in detail the so-called looping case of Mozes's game of numbers, which concerns the (finite) orbits in the reflection representation of affine Weyl groups situated on the boundary of the Tits cone. We give a simple proof that all configurations in the orbit are obtainable from each other by playing the numbers game, and give a strategy for going from one configuration to another. The strategy gives rise to a partition of the finite Weyl group into finitely many graded posets, one for each extending vertex of the associated extended Dynkin diagram. These are selfdual and mutually isomorphic, and dual to the triangulation of the unit hypercube by reflecting hyperplanes, studied by many authors. Unlike the weak and Bruhat orders, the top degree is cubic in the number of vertices of the graph. We explicitly compute the Hilbert polynomial of the poset.