Related papers: Weighted quivers
A mutation loop of a valued quiver $Q$, is a combination of quiver automorphisms (permutations of vertices and valuations) and mutations that sends $Q$ to itself. In this article we study what we called \emph{global mutations loops} which…
A cyclically ordered quiver is a quiver endowed with an additional structure of a cyclic ordering of its vertices. This structure, which naturally arises in many important applications, gives rise to new powerful mutation invariants.
We prove $\textsf{NP-hardness}$ results for determining whether quivers are mutation equivalent to quivers with given properties. Specifically, determining whether a quiver is mutation-equivalent to a quiver with exactly $k$ arrows between…
Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we…
We classify the connected quivers with the property that all the quivers in their mutation class have the same number of arrows. These are the ones having at most two vertices, or the ones arising from triangulations of marked bordered…
Let $Q$ be a rank 3 mutation-cyclic quiver. It is known that every $\mathbf{c}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum_{i=1}^3 x_i^2 + \sum_{1\leq i<j\leq 3} \pm q_{ij} x_i x_j =1,$$where $q_{ij}$ is the…
The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. GHWs are of great interest in many applications since they convey detailed information of linear codes. In this paper, we continue the work of [10] to study…
Introduced in arXiv:2211.12606, biquandle arrow weight invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian…
In this paper, first we introduce a quantity called a partition function for a quiver mutation sequence. The partition function is a generating function whose weight is a $q$-binomial associated with each mutation. Then, we show that the…
We use weighted unfoldings of quivers to provide a categorification of mutations of quivers of types $I_2(2n)$, thus extending the construction of categorifications of mutations of quivers to all finite types.
We define the notion of a weighted unfolding of quivers with real weights, and use this to provide a categorification of mutations of quivers of finite types $H_4$, $H_3$ and $I_2(2n+1)$. In particular, the (un)folding induces a semiring…
Let $(Q,W)$ be a quiver with a non degenerate potential. We give a new description of the \textbf{c}-vectors of $Q$. We use it to show that, if $Q$ is mutation equivalent to a Dynkin quiver, then the set of positive $\mathbf{c}$-vectors of…
We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrizable matrix has a geometric realization by reflections. We also explore the structure of…
We define a functor which gives the "global rank of a quiver representation" and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other "rank functors" for a…
We introduce weighted cycles on weaves of general Dynkin types and define a skew-symmetrizable intersection pairing between weighted cycles. We prove that weighted cycles on a weave form a Laurent polynomial algebra and construct a…
Let $C$ be a simply laced generalized Cartan matrix. Given an element $b$ of the generalized braid semigroup related to $C$, we construct a collection of mutation-equivalent quivers with potentials. A quiver with potential in such a…
Recently, Ringel introduced the resolution quiver for a connected Nakayama algebra. It is known that each connected component of the resolution quiver has a unique cycle. We prove that all cycles in the resolution quiver are of the same…
Quiver mutation plays a crucial role in the definition of cluster algebras by Fomin and Zelevinsky. It induces an equivalence relation on the set of all quivers without loops and two-cycles. A quiver is called mutation-acyclic if it is…
The set of forks is a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let $Q$ be a fork with $n$ vertices, and $\boldsymbol{w}$ be a…
For a quiver with weighted arrows we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented…