Related papers: Associahedra as moment polytopes

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…

We construct compact polyhedra with $m$-gonal faces whose links are generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type $P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of type $A_2$.

Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the constructions…

We define higher polyhedral K-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodin's theory and Quillen's + construction are developed. In the special case of…

We utilize the coherent-constructible correspondence to construct full strongly exceptional collections of nef line bundles in the derived category of a toric variety through the combinatorics of constructible sheaves built from polytopes.…

We show that the mesh mutations are the minimal relations among the $\boldsymbol{g}$-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…

We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the cluster-tilting objects form one equivalence class under mutation.

Let $K$ be a closed convex polyhedron defined by a finite number of linear inequalities. In this paper we refine the theory of abstract tubes (Naiman and Wynn, 1997) associated with $K$ when $K$ is perturbed. In particular, we focus on the…

We give a topological solution to the $\Ainf$ Deligne conjecture using associahedra and cyclohedra. For this we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes…

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with…

We explicitly describe the tropicalization of a type C cluster variety by identifying it with the space of axially symmetric phylogenetic trees. We also study the signed tropicalizations of this cluster variety, realizing them as subfans of…

The construction of a simplicial complex given by polyhedral joins (introduced by Anton Ayzenberg), generalizes Bahri, Bendersky, Cohen and Gitler's $J$-construction and simplicial wedge construction. This article gives a cohomological…

We provide a systematic derivation of cluster alphabets of finite types. The construction is based on a geometric realization of the generalized worldsheets by gluing and folding a pair of polygons. The cross ratios of the worldsheet z…

We reinterpret and generalize the construction of local Shimura varieties and their non-minuscule analogs by viewing them as moduli spaces of admissible pairs. Our main application is a bi-analytic Ax-Lindemann theorem comparing, in the…

We point out that there exists a generalization of instanton symmmetry in the Coulomb phase of 5d nonabelian gauge theories which is capable of measuring a wider class of topological charges of monopole strings. The symmetry is invertible…

Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra \(H\) using central idempotents in right coideal subalgebras and show that any…

We define a class of groups based on parallel computations by pushdown automata. This class generalizes automatic groups. It includes the fundamental groups of all 3-manifolds which obey Thurston' s geometrization conjecture. It also…

We present a construction of chiral and regular polyhedra from subgroups of the general affine group AGL(1,q) for odd prime powers q. In particular, we show that the full group AGL(1,q) occurs as the automorphism group of a chiral…

We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized $F$-structures on manifolds that are not necessarily global products. As applications we characterize invariant generalized…

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…