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→Cluster algebras of rank 3: variables |
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:<math>\{\frac{(1+x_2)x_1+(1+x_2)x_3}{x_1 x_2 x_3},\frac{(1+x_2)x_1+x_3}{x_2 x_3},\frac{1+x_2}{x_3} \}</math>
:<math>\frac{1+x_2}{x_1},\frac{x_1 + x_3}{x_2},\frac{1+x_2}{x_3}, \frac{x_1+(1+x_2)x_3}{x_1x_2}, \frac{(1+x_2)x_1+x_3}{x_2 x_3}, \frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3}</math>.
They correspond to the 6 positive roots of the Dynkin diagram A<sub>3</sub>: more precisely the denominators are monomials in ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, corresponding to the expression of positive roots as the sum of simple roots.
The 3+6 cluster variables generate a cluster algebra of finite type, associated with the Dynkin diagram A<sub>3</sub>.
The 14 clusters are the vertices of the cluster graph, which is an [[associahedron]].
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