Unistructurality of cluster algebras from unpunctured surfaces

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

“…To prove that the type cone 𝕋ℂ( ) is simplicial, we just need to identify which pairs of adjacent maximal cones of  correspond to the facets of 𝕋ℂ( ) and to show that the corresponding linear dependences among their rays positively span the linear dependence among the rays of any pair of adjacent maximal cones of  . When applied to the g -vector fans of cluster algebras (see Subsection 2.2), the type cone approach yields all polytopal realizations of the g -vector fans and thus efficiently revisits and extends the construction of [17]. This new perspective has several advantages, as our proof uniformly applies in the following generality.…”

“…This construction provides a large degree of freedom in the choice of the parameters defining these affine subspaces, and actually produces all polytopes whose normal fan is affinely equivalent to that of Loday's associahedron [60] (see Subsection 2.1). These realizations were then extended by Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas, and Yıldırım [17] in the context of finite-type cluster algebras using tools from representation theory of quivers. More precisely, they fix a finite-type cluster algebra  and consider the real euclidean space ℝ  indexed by the set  of cluster variables of .…”

“…• Any initial seed, regardless of whether it is acyclic or not (see Example 2.18 and Figures 6 to 8 for examples in types 𝐴 3 and 𝐶 3 cyclic). In contrast, the proof of [17] only treats acyclic initial seeds although the approach has since been extended to any seed in [16] (as previously announced by Thomas in a lecture given at RIMS in June 2019). • All finite-type cluster algebras, regardless of whether it is simply laced or not (see Example 2.18 for an example in type 𝐶 3 ).…”

“…• All finite-type cluster algebras, regardless of whether it is simply laced or not (see Example 2.18 for an example in type 𝐶 3 ). In contrast, even with the acyclicity restriction, the result of [17] is first proved in simply laced cases (𝐴, 𝐷, and 𝐸) using representation theory of quivers. Extension to the non simply laced cases can then be argued by a folding argument.…”

“…• Any positive real-valued parameters, regardless of whether they are rational or not. In contrast, the proof of [17] naturally applies to rational parameters and then requires an approximation argument.…”