Related papers: Perfect matching modules, dimer partition function…
It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We…
The monopole-dimer model introduced recently is an exactly-solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a…
We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning…
The perfect matching complex of a simple graph $G$ is a simplicial complex having facets (maximal faces) as the perfect matchings of $G$. This article discusses the perfect matching complex of polygonal line tilings and the $\left(2 \times…
In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property…
This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a…
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of…
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived…
In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the…
Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial…
In this PhD thesis we develop the frame work of triple crossing diagram maps (TCD maps), which describes constrained configurations of points in projective spaces and discrete dynamics on these configurations. We are able to capture the…
We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra $\mathcal{H}(\mathcal{E})$ of…
We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,\mu}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$…
We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely…
Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial…
Building on work of Derksen-Fei and Plamondon, we formulate a conjectural correspondence between additive and monoidal categorifications of cluster algebras, which reveals a new connection between the additive reachability conjecture and…
We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of…
In this note, we consider the $d$-cluster-tilted algebras, the endomorphism algebras of $d$-cluster-tilting objects in $d$-cluster categories. We show that a tilting module over such an algebra lifts to a $d$-cluster-tilting object in this…
Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $\Lambda=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal…
In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting…