
We provide an explicit Dynkin diagrammatic description of the $c$-vectors and the $d$-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types $A_n$ and $D_n$, then we apply the folding method to $D_{n+1}$ and $A_{2n-1}$ to obtain types $B_n$ and $C_n$. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of $c$-vectors for a general cluster algebra.
Rings and Algebras (math.RA), FOS: Mathematics, Mathematics - Combinatorics, Mathematics - Rings and Algebras, Combinatorics (math.CO), Representation Theory (math.RT), 13F60, Mathematics - Representation Theory
Rings and Algebras (math.RA), FOS: Mathematics, Mathematics - Combinatorics, Mathematics - Rings and Algebras, Combinatorics (math.CO), Representation Theory (math.RT), 13F60, Mathematics - Representation Theory
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