Related papers: Positive Fuss-Catalan numbers and Simple-minded sy…

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective…

Obtaining robust galaxy number counts is crucial for understanding galaxy evolution, and submillimetre counts in particular have proven valuable for revising subgrid physics models in cosmological simulations. In confusion-limited surveys,…

We show that the derived wrapped Fukaya category $D^\pi\mathcal{W}(X_{Q}^{d+1})$, the derived compact Fukaya category $D^\pi\mathcal{F}(X_{Q}^{d+1})$ and the cocore disks $L_{Q}$ of the plumbing space $X_{Q}^{d+1}$ form a Calabi--Yau…

We introduce the notion of a Calabi--Yau quadruple as a generalization of Iyama--Yang's Calabi--Yau triple. For each $(d+1)$-Calabi--Yau quadruple, we show that the associated Higgs category is a $d$-Calabi--Yau Frobenius extriangulated…

We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum…

We establish combinatorial interpretations of several identities for the Catalan and Fine numbers and, along the way, we present some new bijections of independent interest. Briefly, we show that C_{n} = 1/(n+1) Sum_{k} (n+1)choose(2k+1)…

In this article, we study the geometric realizations of $m$-cluster categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those four cases, that there is a bijection between $(m+2)$-angulations and isoclasses of basic…

Let $\mathscr{C}$ be a $2$-Calabi-Yau triangulated category with two cluster tilting subcategories $\mathscr{T}$ and $\mathscr{U}$. Results by Demonet-Iyama-Jasso and J{\o}rgensen-Yakimov known as tropical duality says that the index with…

We prove the existence of an $m$-cluster tilting object in a generalized $m$-cluster category which is $(m+1)$-Calabi-Yau and Hom-finite, arising from an $(m+2)$-Calabi-Yau dg algebra. This is a generalization of the result for the ${m =…

In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For…

Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to…

Bijective correspondences are established between (1) silting objects, (2) simple-minded collections, (3) bounded $t$-structures with length heart and (4) bounded co-$t$-structures. These correspondences are shown to commute with mutations.…

We introduce some new Frobenius subcategories of the module category of a preprojective algebra of Dynkin type, and we show that they have a cluster structure in the sense of Buan-Iyama-Reiten-Scott. These categorical cluster structures…

We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster…

The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of…

If $k$ is a field, $A$ a finite dimensional $k$-algebra, then the simple $A$-modules form a simple minded collection in the derived category $\operatorname{D}^b( \operatorname{mod} A )$. Their extension closure is $\operatorname{mod} A$; in…

We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by…

The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains…

Given a generic Looijenga pair $(Y,D)$ together with a toric model $\rho:(Y,D)\rightarrow(\overline{Y},\overline{D})$, one can construct a seed ${\bf s}$ such that the corresponding $\mathcal{X}$-cluster variety $\mathcal{X}_{{\bf s}}$ can…

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers…