Corpus ID: 189927910

# Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

@article{Padrol2019AssociahedraFF,
title={Associahedra for finite type cluster algebras and minimal relations between \$\mathbf\{g\}\$-vectors},
author={Arnau Padrol and Yann Palu and Vincent Pilaud and Pierre-Guy Plamondon},
journal={arXiv: Representation Theory},
year={2019}
}
We show that the mesh mutations are the minimal relations among the $\mathbf{g}$-vectors with respect to any initial seed in any finite type cluster algebra. We then use this algebraic result to derive geometric properties of the $\mathbf{g}$-vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high dimensional positive orthant with well-chosen… Expand
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