Polytopal realizations of finite type $\mathbf{g}$-vector fans

This paper shows that for any finite type initial exchange matrix $\mathrm{B}_\circ$, acyclic or not, there exists a generalized associahedron $\mathsf{Asso}(\mathrm{B}_\circ)$ whose normal fan is the $g$-vector fan $\mathcal{F}^{\mathbf{g}}(\mathrm{B}_\circ)$ of the cluster algebra $\mathcal{A}_{\mathrm{pr}}(\mathrm{B}_\circ)$ with principal coefficients. In particular, when oriented in a suitable direction, the graph of this generalized associahedron $\mathsf{Asso}(\mathrm{B}_\circ)$ is the graph of green mutations of $\mathcal{A}_{\mathrm{pr}}(\mathrm{B}_\circ)$. When $\mathrm{B}_\circ$ is acyclic, $\mathsf{Asso}(\mathrm{B}_\circ)$ coincides with previously known realizations of generalized associahedra, obtained by deleting inequalities in the facet description of the permutahedron. We show that this aspect of the construction remains for any type $A$ initial exchange matrix, but is lost in general for arbitrary finite types. We also prove that the vertex barycenter of all constructed generalized associahedra is at the origin. Finally, our realizations naturally lead to construct a universal associahedron which simultaneously contains all $\mathbf{g}$-vector fans $\mathcal{F}^{\mathbf{g}}(\mathrm{B}_\circ)$ for all initial exchanged matrix of a given finite type.