Related papers: Mutation-acyclic quivers are totally proper
We define a generalized version of the frieze variety introduced by Lee, Li, Mills, Seceleanu and the second author. The generalized frieze variety is an algebraic variety determined by an acyclic quiver and a generic specialization of…
We introduce a deformed squared Markov equation given by $X^2 + Y^2 + Z^2 + (q+q^{-1})(XY+YZ+XZ) = 3(1 + q + q^{-1})XYZ$. Symmetric solutions of this new equation present a remarkable factorization property which allows us to talk about…
We give a precise definition of folded quivers and folded cluster algebras. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite…
This work investigates preserving and reversing unimodality and convexity properties for sequences under transformations defined by sign-regular kernels. It is shown that these transformations only preserve these properties if the kernels…
A $d$-silting object is a silting object whose derived endomorphism algebra has global dimension $d$ or less. We give an equivalent condition, which can be stated in terms of dg quivers, for silting mutations to preserve the $d$-siltingness…
We introduce a new concept of mixed representations of quivers that is a generalization of ordinary representations of quivers and orthogonal (symplectic) representations of symmetric quivers introduced recently by Derksen and Weyman. We…
Reineke and independent other authors proved that every projective variety arises as a quiver Grassmannian. We prove the claim in the title by restricting Reineke's isomorphism to Grassmannians for a fully exact subcategory.
We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for…
We give a geometric proof of the fact that any affine surface with trivial Makar-Limanov invariant has finitely many singular points. We deduce that a complete intersection surface with trivial Makar-Limanov invariant is normal.
The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and…
In this survey paper we give an overview of a generalization, introduced by R. Bautista and the author, of the theory of mutation of quivers with potential developed in 2007 by Derksen-Weyman-Zelevinsky. This new construction allows us to…
Esik and Maletti introduced the notion of a proper semiring and proved that some important (classes of) semirings -- Noetherian semirings, natural numbers -- are proper. Properness matters as the equivalence problem for weighted automata…
To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of…
We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano…
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a…
We construct a special embedding of the translation quiver $\mathbb{Z}Q$ in the three-dimensional affine space $\mathbb{R}^{3}$ where $Q$ is a finite connected acyclic quiver and $\mathbb{Z}Q$ contains a local slice whose quiver is…
It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined…
This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by…
We define an integral form of shifted quantum affine algebras of type $A$ and construct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these…
We observe that under certain conditions on the Lyapunov exponents a semi-invertible cocycle is, indeed, invertible. As a consequence, if a semi-invertible cocycle generated by a H\"{o}lder continuous map $A:M\to M(d, \mathbb{R})$ over a…