Related papers: A determinantal formula for cluster variables in c…
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…
Markov numbers, i.e. positive integers appearing in solutions to $x^2 + y^2 + z^2 = 3xyz$, can be viewed as specializations of cluster variables. The second author and Matsushita gave a generalization of the Markov equation, $x^2 + y^2 +…
As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevisky \cite{[4]}, in this paper, we give the test method of positive…
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…
A given set of data-points in some feature space may be associated with a Schrodinger equation whose potential is determined by the data. This is known to lead to good clustering solutions. Here we extend this approach into a full-fledged…
We study the K_0 and K_1-groups of exact and triangulated categories of perfect complexes, and we apply the results to show how determinant functors on triangulated categories can be used for the construction of Euler characteristics in…
We develop a linked cluster method to calculate the spectral weights of many-particle excitations at zero temperature. The dynamical structure factor is expressed as a sum of exclusive structure factors, each representing contributions from…
We propose a novel graph clustering method guided by additional information on the underlying structure of the clusters (or communities). The problem is formulated as the matching of a graph to a template with smaller dimension, hence…
Graphs are commonly used to represent and visualize causal relations. For a small number of variables, this approach provides a succinct and clear view of the scenario at hand. As the number of variables under study increases, the graphical…
Spectral clustering has become a popular technique due to its high performance in many contexts. It comprises three main steps: create a similarity graph between N objects to cluster, compute the first k eigenvectors of its Laplacian matrix…
Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $\Sigma$ with certain…
We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient…
We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skew-symmetric case, these cluster…
We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of \cite{Palu}) and satisfies…
As data sets continue to grow in size and complexity, effective and efficient techniques are needed to target important features in the variable space. Many of the variable selection techniques that are commonly used alongside clustering…
We consider the problem of model-based clustering in the presence of many correlated, mixed continuous and discrete variables, some of which may have missing values. Discrete variables are treated with a latent continuous variable approach…
Let $Q$ be the affine quiver of type $\widetilde{A}_{2n-1,1}$ and $\mathcal{A}_{q}(Q)$ be the quantum cluster algebra associated to the valued quiver $(Q,(2,2,\dots,2))$. We prove some cluster multiplication formulas, and deduce that the…
We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on…
We investigate skein relations in cluster algebras from punctured surfaces, extending the work of \c{C}anak\c{c}i-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by…
Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…