Abstract
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.
Highlights
1.1 BackgroundFor a given skew-symmetrizable integer matrix B, let A(B) be the cluster algebra with principal coefficients whose initial exchange matrix is B [26, 28]
We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors of any cluster algebra of finite type with principal coefficients and any initial exchange matrix
For any skew-symmetrizable matrix B of cluster finite type, we present the Cartan matrix A(B) as a Dynkin diagram X(B) in the usual way following [34]
Summary
For a given skew-symmetrizable integer matrix B, let A(B) be the cluster algebra with principal coefficients whose initial exchange matrix is B [26, 28]. When B is skew-symmetric, thanks to Kac’s theorem [32], it is enough to prove that the vectors (or their negatives) are identified with the dimension vectors of some indecomposable modules of the path algebra kQ(B) for the quiver Q(B) corresponding to B. This is a common method of proving many known cases. In spite of this beautiful and complete, representation-theoretic description of c- and d-vectors for finite type, little is known about their explicit form, except for type An [12, 44, 53]. It is our hope that the lists presented here will be useful for studying cluster algebras, as the appendix of [3] is for studying Lie algebras
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