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
13 Citations
Tame algebras have dense $\mathbf{g}$-vector fans
The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tameExpand
Tame Algebras Have Dense g-Vector Fans
The $\textbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\textbf{g}$-vectors of its two-term presilting objects. We prove that the $\textbf{g}$-vector fan of a tameExpand
Grothendieck groups and Auslander-Reiten (d + 2)-angles
Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(dExpand
On positive geometries of quartic interactions: one loop integrands from polytopes
Abstract Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. WeExpand
Grothendieck groups in extriangualted categories
The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulatedExpand
Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories
We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has onlyExpand
Stokes polytopes: the positive geometry for ϕ4 interactions
Abstract In a remarkable recent work [1], the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planarExpand
Removahedral Congruences versus Permutree Congruences
The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to allExpand
Positive and negative extensions in extriangulated categories
We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there areExpand
Accordiohedra as positive geometries for generic scalar field theories
We build upon the prior works of [1-3] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction isExpand
...
1
2
...

References

SHOWING 1-10 OF 92 REFERENCES
Brick polytopes, lattice quotients, and Hopf algebras
  • Vincent Pilaud
  • Mathematics, Computer Science
  • J. Comb. Theory, Ser. A
  • 2018
TLDR
This paper describes combinatorially a natural surjection from the permutations to the acyclic $k-triangulations, and uses this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations. Expand
NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON
We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$ -tuples in $[n]$ that was first considered by Petersen et al. [J. Algebra 324(5) (2010),Expand
$\tau$-tilting finite algebras, bricks and $g$-vectors
The class of support τ -tilting modules was introduced recently by Adachi-Iyama-Reiten so as to provide a completion of the class of tilting modules from the point of view of mutations. In thisExpand
Compatibility fans for graphical nested complexes
TLDR
A compatibility degree is defined that the compatibility vectors of all tubes of a graph "$G$ with respect to an arbitrary maximal tubing on $G$ support a complete simplicial fan realizing the nested complex of $G$. Expand
Cluster-tilted algebras are Gorenstein and stably Calabi–Yau
Abstract We prove that in a 2-Calabi–Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We showExpand
Polyhedral models for generalized associahedra via Coxeter elements
Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin, and A. Zelevinsky associated to each finite type root system a simple convex polytope, called generalized associahedron. TheyExpand
Y-systems and generalized associahedra
The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system D, the periodicity conjecture of Al. B. Zamolodchikov [24] that concerns Y-systems, a particular class ofExpand
On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In thisExpand
Cambrian Lattices
For an arbitrary finite Coxeter group W , we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to eachExpand
Permutahedra and generalized associahedra
Given a finite Coxeter system (W,S)(W,S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W , we construct a simple polytope whose outer normal fan is N. Reading'sExpand
...
1
2
3
4
5
...