Questions tagged [cluster-algebras]

Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.

Filter by
Sorted by
Tagged with
2 votes
0 answers
51 views

Finding relations of cluster-tilted algebras

Let $Q$ be a quiver. Then we can use mutation (in the cluster algebra setting) to obtain a new quiver $Q'$ and to each such a quiver $Q'$ there corresponds a unique cluster-tilted algebra, which is a ...
Mare's user avatar
  • 25.7k
2 votes
1 answer
213 views

Grassmannian $\mathrm{Gr}(k, \pm \infty)$ in infinite dimension

$\DeclareMathOperator\Gr{Gr}$The Grassmnnian variety $\Gr(k,n)$ is the set of $k$-dimensional subspaces of $\mathbb{C}^n$. The coordinate ring $\mathbb{C}[\Gr(k,n)]$ is generated by Plucker ...
Jianrong Li's user avatar
  • 6,101
1 vote
0 answers
125 views

Number of cluster variables associated to A type quivers

In a seminar/reading course about cluster algebras, we came across the fact that the number of cluster variables for the cluster algebra associated to mutating the quiver $A_n$ is $n(n+3)/2$ (rather, ...
Andrea B.'s user avatar
  • 315
3 votes
0 answers
76 views

Theta functions in acyclic cluster algebras

Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $...
Antoine de Saint Germain's user avatar
1 vote
0 answers
59 views

When the dg cluster category of a quiver is saturated?

Let $Q$ be a finite quiver without oriented cycles. In https://arxiv.org/abs/0807.1960 , Keller defines the dg cluster category $C_Q$ of $Q$. When is $C_Q$ smooth proper dg-category? If $Q$ is a ...
OOOOOO's user avatar
  • 357
10 votes
1 answer
631 views

Rigorous proof of the pentagon identity

I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra. For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...
Estwald's user avatar
  • 987
4 votes
0 answers
231 views

Road map for learning cluster algebras

I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
It'sMe's user avatar
  • 767
1 vote
0 answers
49 views

Non-vanishing of generalized minors on T-stable unipotent subgroups

Let $G$ be a complex simply connected algebraic group, $T$ a maximal torus of $G$ and $B$, $B^-$ Borel subgroups which are opposite with respect to $T$ and let $U$ (resp. $U^-$) be the unipotent ...
Luca Francone's user avatar
2 votes
0 answers
108 views

What algebraic varieties arise as intersections of finitely many cluster charts?

A cluster variety $V$ admits, by definition, many charts of the form $(\mathbb{C}^*)^n \hookrightarrow V$. These charts do not always cover the variety of interest, but when they do, one could e.g. ...
Vivek Shende's user avatar
  • 8,663
4 votes
0 answers
239 views

A technical question about a paper by Gross-Hacking-Keel

I have a technical question on the commutativity of diagrams (2.11) and (2.12) in the paper "Birational geometry of cluster algebras" by Gross-Hacking-Keel: For the leftmost square in (2.11),...
mikeS's user avatar
  • 61
5 votes
0 answers
171 views

"Cluster algebra" structure for finite distributive lattices

Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets). For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...
Sam Hopkins's user avatar
  • 22.5k
3 votes
0 answers
149 views

The Grassmann twist-map, an associated semi-group action, and RSK

Let me begin by setting some notation: Let $\mathrm{Mat}_{k,n}(\Bbb{R})$ denote the vector space of all $k \times n$ real-valued matrices. Given $g \in \mathrm{Mat}_{k,n}(\Bbb{R})$ and two (ordered) ...
Jeanne Scott's user avatar
  • 1,847
4 votes
1 answer
99 views

How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?

The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...
Richard Chen's user avatar
12 votes
2 answers
306 views

Easy way to understand theta basis for X-cluster algebras of finite type?

For $\mathcal A$-cluster algebras of finite type, it is very easy to describe the theta-basis: it consists of the cluster monomials. Is there any similarly easy way to describe the theta-basis for $\...
Hugh Thomas's user avatar
  • 6,075
0 votes
1 answer
121 views

About cluster variables obtained by (sequentially) mutating at exchangeable variables from an initial cluster

Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a ...
amator2357's user avatar
4 votes
1 answer
349 views

Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces. Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
giulio bullsaver's user avatar
3 votes
0 answers
109 views

Fundamental representation bases and generalized minors

Let $G$ be a simple simpy connected complex algebraic group. I was wondering if there is a clear relationship between the generalized minors (defined by Berenstein, Fomin and Zelevinsky) and bases of ...
F.H.A's user avatar
  • 201
4 votes
0 answers
160 views

Positivity conjecture for Somos sequences

Let $\{s_n\}$ be the Somos-$4$ sequence, which is defined by $$s_{n+4}s_n=\alpha s_{n+3}s_{n+1}+\beta s_n^2.$$ It is known that $s_n$ is a Laurent polynomial: $s_n\in\mathbb{Z}[s_1^{\pm1}, \ldots, s_4^...
Alexey Ustinov's user avatar
13 votes
2 answers
381 views

Quiver representations of type $D_n$ mutation class

I was wondering if there is a classification of the indecomposable quiver representations of (not necessarily acyclic) quivers that are mutation equivalent to the $D_n$ Dynkin diagram. Such quivers ...
Kayla Wright's user avatar
8 votes
1 answer
222 views

Grassmannian cluster algebra of infinite type has no trees in its mutation class

The question is why the statement in the title is true (is it?). To elaborate, recall that Grassmannian cluster algebra, according to Scott`s paper Grassmannians and Cluster Algebras, is the cluster ...
Andrei Smolensky's user avatar
1 vote
0 answers
81 views

Defining cluster algebras of finite type $\mathrm{A}$ by generators and relations

Consider a cluster algebra of finite type $\mathrm{A}$. The set of all (distinct) cluster variables is of finite cardinality, denote it by $k$, for such algebra. Is it true that, for an arbitrary ...
amator2357's user avatar
3 votes
0 answers
96 views

Geometric description of a type $A$ cluster algebra with universal coefficients

In Thm. 12.4 in Fomin and Zelevinsky - Cluster algebras IV: Coefficients we are given a recipe for constructing a cluster algebra with universal coefficients. The recipe is given in terms of (almost ...
amator2357's user avatar
2 votes
0 answers
74 views

Berenstein-Fomin-Zelevinsky's Ininital seeds and initial seeds from Postnikov diagrams

In Cluster algebra III by Berenstein-Fomin-Zelevinsky, Theorem 2.10, for any pair of reduced words $(u,v)$, they constructed an initial seed for the cluster algebra $\mathbb{C}[B^{u,v}]$, where $B^{u,...
Jianrong Li's user avatar
  • 6,101
6 votes
1 answer
274 views

Is this Laurent phenomenon explained by invariance/periodicity?

In Chapter 4 (page 23, subsection "Somos sequence update") of his Tracking the Automatic Ant, David Gale discusses three families of recursively defined sequences of numbers, all due to Dana ...
darij grinberg's user avatar
4 votes
0 answers
63 views

Why does this cluster tilting object form a local slice?

I am reading the paper "Cluster automrphisms", here is the link: http://prospero.dmat.usherbrooke.ca/ibrahim/publications/Cluster_Automorphisms.pdf In the proof of lemma 3.1 I am stuck: For ...
Xiaosong Peng's user avatar
2 votes
0 answers
60 views

Multiplication formula in Grassmannian cluster categories

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-...
Jianrong Li's user avatar
  • 6,101
17 votes
4 answers
2k views

Some interesting and elementary topics with connections to the representation theory?

I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (...
kotlinski's user avatar
  • 181
2 votes
1 answer
187 views

Choice of a ground ring for cluster algebras

In order to define cluster algebra one needs to define its ground ring. In most cases, we take a group $P$ (often called a coefficient group) which is taken to be an abelian multiplicative group . ...
amator2357's user avatar
1 vote
1 answer
243 views

Cluster algebras and tropical points

Recently, I have been looking at some articles about bases for cluster algebras and came across the idea of tropical points. I should highlight here that unfortunately I have no background on ...
amator2357's user avatar
2 votes
0 answers
64 views

Piecewise linear $\sigma_i$ - notation question

In cluster algebra framework, in order to get root clusters, a modified version of a simple reflection is used. Define $\sigma_i:\Phi_{\geq -1} \to \Phi_{\geq -1}$ by setting: $ \sigma_i(\alpha) = \...
amator2357's user avatar
19 votes
2 answers
1k views

Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?

Gross-Hacking-Keel-Kontsevich (https://arxiv.org/abs/1411.1394) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain ...
Sam Hopkins's user avatar
  • 22.5k
11 votes
0 answers
399 views

Scattering diagram for the cluster algebra $ \mathbb C [N]$

Five years ago, Gross-Hacking-Keel-Kontsevich made a major advance in the theory of cluster algebras, by constructing bases of cluster algebras in large generality. A key tool in their construction ...
Joel Kamnitzer's user avatar
1 vote
1 answer
198 views

Are there some relations between F-polynomials and theta functions?

F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to ...
Jianrong Li's user avatar
  • 6,101
3 votes
0 answers
97 views

Does singularity confinement imply a fixed pattern of irreducible factors?

Consider a rational map $f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...
Sam Hopkins's user avatar
  • 22.5k
13 votes
2 answers
575 views

Integer but not Laurent sequences

Are there any sequence given by a recurrence relation: $x_{n+t}=P(x_t,\cdots,x_{t+n-1})$, where $P$ is a positive Laurent Polynomial, satisfy: if $x_0=\cdots=x_{n-1}=1$, then the sequence is only ...
Sylvester W. Zhang's user avatar
3 votes
1 answer
160 views

"Solution" of finite cluster algebras

Consider the cluster algebras $A_n$ and $D_n$. Choose any cluster $x$, is there an explicit formula that express all other cluster variables in terms of $x$?
giulio bullsaver's user avatar
8 votes
0 answers
128 views

"Cross-Ratios" for D_n cluster algebra

Both cluster algebras $A_n$ and $D_n$ admit an interpretation in terms of (tagged) triangulations of Riemann surfaces - respectively a Poincaré disk with n+3 punctures on the boundary and a Poincaré ...
giulio bullsaver's user avatar
11 votes
0 answers
255 views

Cluster algebra and Fenchel Nielsen coordinates

Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
giulio bullsaver's user avatar
2 votes
0 answers
53 views

Differential (in)dependancies between cluster variables

Let $ \mathcal A$ be the complex cluster algebra obtained from the initial seed $(a,x, B)$ where : $x$ stands for a $n$-tuple $(x_1,\ldots,x_n)$ of complex indeterminates, idem for $a=(a_1,\ldots,...
Elbabak's user avatar
  • 347
29 votes
2 answers
1k views

Determining if a rational function has a subtraction-free expression

This question was first asked by Mehtaab Sawhney in Alex Postnikov's combinatorics class. Given a rational function $F=P(x_1,...,x_n)/Q(x_1,...,x_n)$ with (say) integer coefficients, it is often of ...
Christian Gaetz's user avatar
6 votes
0 answers
79 views

Non-rigid indecomposable summands of simple-minded collections in bounded derived category of hereditary algebras

Let $\Lambda$ be a hereditary algebra over an algebraically closed field $k$. Let $S$ be one of the indecomposable summands of one simple-minded collection in $D^b(\Lambda)$. Is it true that $S$ is ...
Ying Zhou's user avatar
  • 417
1 vote
1 answer
212 views

Why do finitely many cluster variables imply finitely many y-variables?

Suppose we have a seed $(x,y,B)$ where $B$ is a skew-symmetrizable matrix, $x = \{x_1,\ldots,x_n\}$ and y is an n-tuple of elements in $Trop\{x_{n+1},\ldots,x_m\}$. If there are finitely many cluster ...
user118668's user avatar
1 vote
0 answers
163 views

(b,c) rank 2 cluster algebras

Let $x$ and $y$ be variables. Consider the following recurrence: \begin{equation} u_{n}:= \begin{cases} \displaystyle{\frac{1+u_{n-1}^b}{u_{n-2}}} & if\ n\ \text{is even},\\ &\\ \displaystyle{\...
Hector Blandin's user avatar
0 votes
0 answers
177 views

Automorphisms of weighted quiver

I am reading this paper strongly primitiv species with potentials I : mutations. In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...
Xiaosong Peng's user avatar
4 votes
1 answer
124 views

Geometric realizations of cluster categories of non-simply-laced types

Geometric realizations of cluster categories of simply-laced types are studied in the following papers. Philippe Caldero, Frédéric Chapoton, and Ralf. Schiffler, Quivers with relations arising from ...
Jianrong Li's user avatar
  • 6,101
6 votes
1 answer
289 views

What is the status of a problem about cluster categories?

Let $H$ be a hereditary algebra of Dynkin type. There is a cluster category $\mathcal{C}_H$ defined by Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov in Tilting ...
Jianrong Li's user avatar
  • 6,101
15 votes
1 answer
1k views

Applications of cluster algebras

Why are so many algebraists nowadays interested in cluster algebras? (This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)
ThiKu's user avatar
  • 10.2k
2 votes
0 answers
67 views

Question about the mutation of a cluster seed associated to any word of the braid semigroup

Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ ...
Daisy's user avatar
  • 338
2 votes
0 answers
132 views

Questions about cluster $\mathcal{X}$-varieties and amalgamation

I am trying to learn the amalgamation of two cluster seeds and I am reading the paper https://arxiv.org/pdf/math/0508408.pdf written by V.V. Fock and A. B. Goncharov. I am at a loss for the Lemma 2....
Daisy's user avatar
  • 338
6 votes
1 answer
128 views

Decidability and Cluster algebras

Recall the definition of a cluster algebra, which can be seen as a (possibly infinite) graph, where each vertex is a tuple of a quiver and Laurent expressions at some of the vertices of the quiver. ...
Per Alexandersson's user avatar