Questions tagged [cluster-algebras]
Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.
87
questions
4
votes
0answers
105 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^...
12
votes
2answers
188 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 ...
6
votes
1answer
113 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 ...
1
vote
0answers
57 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 ...
3
votes
0answers
65 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 ...
2
votes
0answers
61 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,...
5
votes
1answer
153 views
Is this Laurent phenomenon explained by invariance/periodicity?
In Chapter 4 of his Tracking the Automatic
Ant, David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana Scott and inspired by the Somos sequences:
Sequence 1. ...
3
votes
0answers
52 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 ...
2
votes
0answers
46 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-...
15
votes
4answers
1k 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 (...
2
votes
1answer
127 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 . ...
1
vote
1answer
138 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 ...
2
votes
0answers
54 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) =
\...
18
votes
2answers
851 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 ...
11
votes
0answers
276 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 ...
1
vote
1answer
160 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 ...
3
votes
0answers
92 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 ...
13
votes
2answers
539 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 ...
3
votes
1answer
143 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 $x$?
8
votes
0answers
114 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 ...
11
votes
0answers
168 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 ...
2
votes
0answers
51 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,...
27
votes
2answers
943 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 ...
6
votes
0answers
72 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 ...
1
vote
1answer
181 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 ...
1
vote
0answers
154 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{\...
0
votes
0answers
117 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-...
4
votes
1answer
119 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 ...
6
votes
1answer
252 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 ...
13
votes
1answer
951 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.)
2
votes
0answers
55 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 $\prod$ ...
2
votes
0answers
120 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....
6
votes
1answer
101 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. ...
8
votes
2answers
544 views
A question about the quivers with potentials
Let $Q=(Q_0,Q_1,h,t)$ be a quiver consisted of a pair of finite sets $Q_0$(vectors),and $Q_1$ (arrows) supplied with two maps $h : Q_1 → Q_0$ (head) and $t : Q_1 → Q_0$ (tail ). This definition allows ...
3
votes
0answers
192 views
Definition of loop amplituhedrons
In the paper The Amplituhedron
, Nima Arkani-Hamed and Jaroslav Trnka introduced the geometric object amplituhedron. It is defined as follows (see also the lecture notes).
Let $Z$ be a $(k+m)\times ...
2
votes
0answers
94 views
Weighted quiver in Keller's Java Quiver mutation [closed]
I need to do mutation using weighted quivers. There are weights on the arrows which looks like:
How to draw weighted quiver in Keller's Java Quiver mutation:
https://webusers.imj-prg.fr/~bernhard....
4
votes
1answer
95 views
Rigid regular objects of path algebras of tame quivers
In the paper On Maximal Green Sequences by Brustle, Dupont and Perotin the authors argued that in a path algebra $\Lambda=kQ$ of a tame quiver $Q$ with $n$ vertices each tilting module contains at ...
3
votes
1answer
511 views
What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?
I am reading the lecture notes and would like to know more about canonical basis.
What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?
...
13
votes
1answer
420 views
Is there some relation between cluster algebras and crystal graphs?
Cluster algebras are closely related to totally positivity in algebraic groups and canonical bases in quantum groups.
Is there some relation between cluster algebras and crystal graphs? Can the ...
2
votes
1answer
148 views
Which cluster algebras where the existence of maximal green sequences is still unknown?
Maximal green sequences are studied in many papers. For example, Maximal Green Sequences for Cluster Algebras Associated to the n-Torus by Eric Bucher, On Maximal Green Sequences by Thomas Brüstle, ...
3
votes
1answer
127 views
Reference request: coordinate ring of $OSP(2p|n)$
In the paper, the orthosymplectic supergroup $OSP(2p|n)$ is defined as follows.
Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are ...
3
votes
2answers
210 views
How to translate cluster X-coordinates to cluster A-coordinates?
In the paper, (5.28) on page 35 is a formula which translate cluster A-coordinates to cluster X-coordinates:
\begin{align}
x_i = \prod_{i \to j} a_j.
\end{align}
Is there a formula which translate ...
2
votes
0answers
111 views
Shear coordinates, lambda lengths, cluster variables
I am trying to understand the relations among Shear coordinates, lambda lengths, cluster variables, in the paper. Is the following correct?
Lambda lengths = cluster A-variables
Shear coordinates = ...
4
votes
2answers
221 views
References about tropical cluster algebras and tropical Laurent phenomenon
Are there some references about tropical cluster algebras and tropical Laurent phenomenon? I searched on Google but only found one paper: Tropical Plucker functions and their bases
.
Thank you very ...
3
votes
1answer
181 views
Trying to understand the proof of Laurent phenomenon of cluster algebras
I am trying to understand the proof of Laurent phenomenon of cluster algebras in the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras. Chapters 1-3, arXiv:1608....
4
votes
2answers
331 views
Reference request: Associahedron
I am reading Sergey Fomin's and Nathan Reading's paper Root Systems and Generalized Associahedra.
I need a good reference for associahedron of classical types. Besides, whether there are some ...
0
votes
2answers
144 views
Tropical version of exchange relations in cluster algebras
The exchange relation in a cluster algebra is
\begin{align}
x_k' = \frac{1}{x_k} (\prod_{j \to k}x_j + \prod_{k \to j} x_j).
\end{align}
Do we have some tropical version of this relation? Are there ...
0
votes
1answer
205 views
What are the cluster algebra structures on $Gr(3,5)$?
In the paper, cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have ...
1
vote
1answer
60 views
A question about exchange pattern
Exchange pattern, see Section 2 in "cluster algebras I: foundations" by Fomin and Zelevinsky or How to understand exchange pattern?
Given an example $\cdots \overset{2}{-} t_1 \overset{1}{-} t_2 \...
3
votes
1answer
152 views
How to understand exchange pattern?
I am reading an paper "cluster algebras I: foundations" by Fomin and Zelevinsky.
Let $I = \{1,2, \ldots, n\}$ and $\mathbf{x}$ a cluster.
For each $t \in \mathbb{T}_n$, let $\mathbf{x}(t) = (x_i(t))...
