Folded galleries and moment graphs

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.

Figures (6)

• Figure 1: On the left: Coxeter complex of $\tilde{\mathtt{A}}_2$ with folded galleries. On the right: Bruhat moment graph of $\mathtt{A}_2$.
• Figure 2: Examples of minimal galleries satisfying the three conditions of \ref{['lemma:CharacterizationOfMinimalityOfGalleriesByCrossings']} with respect to the root $\alpha_1$. The blue arc indicates the orientation defining Weyl chamber.
• Figure 3: Visualization of \ref{['lemma:CrossingDirectionsMinimalGalleries']}. The light blue circle indicates the boundary sphere $\partial \Sigma$. The depicted gallery $\gamma$ ends in $\mathcal{C}_v$. Since $v \in \partial \Sigma$ is on the $\partial \phi_w$-negative side of the horizontal hyperplane $\partial {H}$, crossings of horizontal hyperplanes of $\gamma$ in $\Sigma$ are positive. Since $\partial v$ is on the $\partial \phi_w$-positive side of the boundary hyperplane corresponding to hyperplanes in $\Sigma$ perpendicular to $\alpha_1$, all crossings of $\alpha_1$-hyperplanes of $\gamma$ are $\phi_w$-negative.
• Figure 4: Folded galleries.
• Figure 5: On the left: Coxeter complex of $\widetilde{\mathtt{B}}_2$ with folded galleries. On the right: Bruhat moment graph of $\mathtt{B}_2$.

Theorems & Definitions (48)

• Theorem
• Definition 2.1: Spherical direction
• Definition 2.2: Parallelism class of a hyperplane
• Definition 2.3: $i$-strip
• Definition 2.4: Local Weyl chamber
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Definition 3.1
• Definition 3.2: Concatenation of galleries