Related papers: Dyck Paths, Configuration Spaces and Polytopes For…
This paper surveys recent contructions in higher Auslander--Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type…
We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic…
Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake…
Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321-avoiding bijections via the Billey-Jockusch-Stanley bijection. Thus to every 321-avoiding permutation $\pi$ we…
In this article we investigate the lattices of Dyck paths of type $A$ and $B$ under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with…
Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a…
This present paper is devoted to the study of a class of Nakayama algebras $N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$ subject to the nilpotency degree $r$ for each sequence of $r$ consecutive arrows. We…
We give a combinatorial classification of Nakayama algebras of small homological dimension using the Krattenthaler bijection between Dyck paths and 132-avoiding permutations.
We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with…
We give a complete classification of all $d$-representation-finite symmetric Nakayama algebras and of all $d$-representation-finite trivial extensions of path algebras of quivers, over an arbitrary field. As a consequence we get a…
Adapting a recent work of Brannan et al., on extending graph $C^*$-algebras to Quantum graphs, we introduce "Quantum Quivers" as an analogue of quivers where the edge and vertex set has been replaced by a $C^*$-algebra and the maps between…
We present combinatorial bijections and identities between certain skew Young tableaux, Dyck paths, triangulations, and dissections.
We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional…
We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \kk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted…
In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we…
Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are…
We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations…
In this paper, we give part-preserving bijections between three fundamental families of objects that serve as natural framework for many problems in enumerative combinatorics. Specifically, we consider compositions, Dyck paths, and…
We study the moduli space of twisted quasimaps from a fixed smooth projective curve to a Nakajima's quiver variety and the moduli space of $\delta$-stable framed twisted quiver bundles with moment map relations. We show that they carry…
Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi-Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated…