Related papers: Mutations Vs. Seiberg duality
We propose a duality between quiver gauge theories and the combinatorics of dimer models. The connection is via toric diagrams together with multiplicities associated to points in the diagram (which count multiplicities of fields in the…
The concept of modulation is generalized to pseudo-modulation and its subclasses including pre-modulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called {\em natural…
In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…
Given a finite quiver, its double may be viewed as its non-commutative "cotangent" space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while…
The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give…
Let $G$ be a finite abelian group acting on a path algebra $kQ$ by permuting the vertices and preserving the arrowspans. Let $W$ be a potential on the quiver $Q$ which is fixed by the action. We study the skew group dg algebra $\Gamma_{Q,…
We introduce a new function on the set of pairs of cluster variables via $f$-vectors, which we call it the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility…
$F$-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of $F$-invariant is that it is a coordinate-free invariant, that is, it is mutation…
Stacks of D3-branes placed at the tip of a cone over a del Pezzo surface provide a way of geometrically engineering a small but rich class of gauge/gravity dualities. We develop tools for understanding the resulting quiver gauge theories…
We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its…
We propose a mutation formula for the general rank from a principal component ${\rm PC}(\delta)$ of representations to another one ${\rm PC}(\epsilon)$ for a quiver with potential. We give sufficient conditions for the formula to hold. In…
We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can…
We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that…
For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how…
The quiver Yangians were originally defined for the quiver and superpotential from string theory on general toric Calabi-Yau threefolds, and serve as BPS algebras of these systems. Their characters reproduce the unrefined BPS indices, which…
In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe…
We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in…
Permutation equivariant neural networks are often constructed using tensor powers of $\mathbb{R}^{n}$ as their layer spaces. We show that all of the weight matrices that appear in these neural networks can be obtained from Schur-Weyl…
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…
Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support {\tau}-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given…