Related papers: Leclerc's conjecture on a cluster structure for ty…

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from…

We propose a conjecture on the relative twist formula of $\ell$-adic sheaves, which can be viewed as a generalization of Kato-Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative…

We prove a conjecture of Geiss, Leclerc and Schr\"{o}er, producing cluster algebra structures on multi-homogeneous coordinate ring of partial flag varieties, for the case $G_2$. As a consequence we sharpen the known fact that coordinate…

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…

We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…

We produce a large class of generalized cluster structures on the Drinfeld double of $\text{GL}_n$ that are compatible with Poisson brackets given by Belavin-Drinfeld classification. The resulting construction is compatible with the…

A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points satisfying (i) every two blocks intersect each other in $\lambda$ points for a fixed $\lambda < v$ (ii) there…

We have examined the spatial distribution of substructure in clusters of galaxies using Einstein X-ray observations. Subclusters are found to have a markedly anisotropic distribution that reflects the surrounding matter distribution on…

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable.…

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld…

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$,…

Suppose that $A$ and $B$ are closed subsets of a Euclidean space such that $A\cap B\neq\varnothing$, and we aim to find a point in this intersection with the help of the sequences $(a_n)_\nnn$ and $(b_n)_\nnn$ generated by the \emph{method…

In a survey paper in 2011, Amiot proposed a conjectural characterisation of the cluster categories which were conceived in the mid 2000s to lift the combinatorics of Fomin-Zelevinsky's cluster algebras to the categorical level. This paper…

This is a survey article on Richardson varieties and their combinatorics. A Richardson variety is the intersection, inside the flag manifold GL_n/B_+, of a Schubert cell (B_- u B_+)/B_+ and an opposite Schubert cell (B_+ w B_+)/B_+ (or the…

This paper presents a clustering algorithm that is an extension of the Category Trees algorithm. Category Trees is a clustering method that creates tree structures that branch on category type and not feature. The development in this paper…

First, we show that conjugate Lagrangian fillings, associated to plabic graphs, and Lagrangian fillings obtained as Reeb pinching sequences are both Hamiltonian isotopic to Lagrangian projections of Legendrian weaves. In general, we…

This work draws inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping…

For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $\tau$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras.…

The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings…

We generalize to the non simply-laced case results of Gei\ss, Leclerc and Schr\"oer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non…