The boundary algebra of a GL_m-dimer

Figures & data

Figure 1. Part of the quiver $\Gamma (n).$

Figure 2. α is part of a positive cycle p and a negative cycle q.

Figure 3. Constructing the GL_m-dimer on a triangle. Here m = 4.

Figure 4. A GL₂-dimer of a pentagon.

Figure 5. The white point of the dimer is on the left hand side of the arrow.

Figure 6. Quiver of the GL₂-dimer of a triangle.

Figure 7. $\Gamma (5).$

Figure 8. $Q_F(8):$ The quiver of a fan triangulation of the octagon.

Figure 9. Reduction procedure.

Figure 10. A fan triangulation of the $n+1$-gon and new part of the dimer. The labeling corresponds to the vertices of the $n+1$-gon.

Figure 11. Part of a reduced GL₂-dimer of the $n+1$-gon and the quiver $Q(n+1).$

Figure 12. Diagonal flip of a triangulation.

Figure 13. Flip of the diagonal $(1,j)$ changes the dimer algebra to ${\Lambda }_{\mu Q_F(n)}.$

Figure 14. Effect of an arbitrary flip on arrows of type $z_{2k}.$

Figure 15. Reduced quiver of the GL₅-dimer of a quadrilateral with diagonal incident with 1.

Figure 16. $Q_F(4,6):$ Quiver of the GL₄-dimer of the fan triangulation of a hexagon. For illustration, some arrows are labeled.

Figure 17. $\Gamma (4,6)$ as in Definition 4.6. Arrows x are black, arrows y are red, arrows z are blue.