Folded galleries and moment graphs

TL;DR

This paper explores the combinatorial structure of folding patterns in affine Coxeter complexes, linking them to Bruhat moment graphs to determine the direction of folded galleries.

Contribution

It introduces a new characterization of folding patterns using Bruhat moment graphs and provides a method to determine the end alcove's direction in positively folded galleries.

Findings

Folding patterns are characterized via Bruhat moment graphs.

A method to determine the spherical direction of the end alcove is established.

The connection simplifies understanding of affine Coxeter complex galleries.

Abstract

We characterize folding patterns, the combinatorial options of folding minimal alcove-to-alcove galleries in affine Coxeter complexes positively with respect to Weyl chamber orientations of the Coxeter complex, by drawing a connection to the Bruhat moment graph of the associated spherical Coxeter group. We also prove how to determine the spherical direction of the end alcove of a positively folded gallery using these graphs.