Related papers: Universal quivers

Let $\Gamma^{n}$ be the cone of an $(n-1)$-complete algebra over an algebraically closed field $k$. In this paper, we prove that if the bound quiver $(Q_{n},\rho_{n})$ of $\Gamma^{n}$ is a truncation from the bound McKay quiver…

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…

It is well-known that a quiver Q of type A_n is representation-finite, and that its indecomposable representations are thin (all Jordan-Hoelder multiplicities are 0 or 1). By now, various methods of proof are known. The aim of this note is…

In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the…

We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space…

Let k a characteristic zero field. We give a characterization for the finite quiver k-algebras, based on double derivations. More precisely, we prove that if an associative and unitary k-algebra have a family of double derivations…

Let $\mathbb{P}_n$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. In this paper, we prove the following result: Let $\mathbb{I}$ be the unit tensor of order $m\ge3$ and dimension…

We prove a multiplication theorem of a quantum Caldero-Chapoton map associated to valued quivers which extends the results in \cite{DX}\cite{D}. As an application, when $Q$ is a valued quiver of finite type or rank 2, we obtain that the…

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular…

Let A be a basic finite dimensional and connected algebra over an algebraically closed field k with zero characteristic. If the ordinary quiver of A has no double bypasses, we show that A admits a Galois covering which satisfies a universal…

We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.

Let $M = (m_{ij})$ be an $n \times n$ square matrix of integers. For our purposes, we can assume without loss of generality that $M$ is homogeneous and that the entries are non-increasing going leftward and downward. Let $d$ be the sum of…

Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant…

A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices…

We prove that every planar straight line graph with $n$ vertices has a conforming quadrilateral mesh with $O(n^2)$ elements, all angles $\leq 120^\circ$ and all new angles $\geq 60^\circ$. Both the complexity and the angle bounds are sharp.…

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for…

In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2}…

We show that for every m in N, there exists an n in N such that every embedding of the complete graph K_n in R^3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r in N such that every…

In this paper, we associate a quiver with superpotential to each $d$-angulation of a (unpunctured) marked surface. We show that, under quasi-isomorphisms, the flip of a $d$-angulation is compatible with Oppermann's mutation of (the Ginzburg…

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of…