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
8 votes
2 answers
923 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 ...
Daisy's user avatar
  • 338
3 votes
0 answers
218 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 ...
Jianrong Li's user avatar
  • 6,101
2 votes
0 answers
124 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....
Jianrong Li's user avatar
  • 6,101
4 votes
1 answer
117 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 ...
Ying Zhou's user avatar
  • 417
3 votes
1 answer
932 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? ...
Jianrong Li's user avatar
  • 6,101
14 votes
1 answer
533 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 ...
Jianrong Li's user avatar
  • 6,101
2 votes
1 answer
170 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, ...
Jianrong Li's user avatar
  • 6,101
3 votes
1 answer
158 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 ...
Jianrong Li's user avatar
  • 6,101
3 votes
2 answers
232 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 ...
Jianrong Li's user avatar
  • 6,101
2 votes
0 answers
139 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 = ...
Jianrong Li's user avatar
  • 6,101
4 votes
2 answers
241 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 ...
Jianrong Li's user avatar
  • 6,101
3 votes
1 answer
300 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....
Jianrong Li's user avatar
  • 6,101
4 votes
2 answers
362 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 ...
bing's user avatar
  • 331
0 votes
2 answers
162 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 ...
Jianrong Li's user avatar
  • 6,101
0 votes
1 answer
253 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 ...
Jianrong Li's user avatar
  • 6,101
1 vote
1 answer
71 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 \...
bing's user avatar
  • 331
3 votes
1 answer
167 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))...
bing's user avatar
  • 331
1 vote
1 answer
104 views

Softwares which compute all non-isomorphic quivers in a mutation class

Let $Q$ be a quiver. The mutation class of $Q$ consists of all quivers which can be obtained from $Q$ by a sequence of mutations. Are there some softwares which compute all non-isomorphic quivers in a ...
Jianrong Li's user avatar
  • 6,101
0 votes
1 answer
182 views

Mutation equivalence of quivers

Given two orientations $Q, Q'$ of a Dyinkin diagram. Is it always true that after a sequence of mutations, $Q$ becomes $Q'$? Are the some references about this? Thank you very much.
Jianrong Li's user avatar
  • 6,101
23 votes
0 answers
1k views

Is A276175 integer-only?

The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
uvdose's user avatar
  • 593
1 vote
1 answer
200 views

Mutation of valued quivers

Mutations of valued quivers are defined in cluster algebras II, Proposition 8.1 on page 28. I have a question about the number $c'$. For example, let $a = 2, b=1, c=1$ and consider the quiver $Q$: $1 ...
Jianrong Li's user avatar
  • 6,101
4 votes
0 answers
237 views

polynomials satisfying the Plücker relation

Let $S_{12}$, $S_{13}$, $S_{14}$, $S_{23}$, $S_{24}$, $S_{34}$ be complex homogeneous polynomials in 4 variables satisfying the Plücker relation : $$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$ ...
Hephaistos's user avatar
1 vote
1 answer
103 views

How to draw a quiver for a pseudoline arragement?

In the lecture notes, on page 24, there is an example of drawing a quiver for a pseudoline arragement. What is the rule to draw a quiver for a pseudoline arragement? I don't know how to put the ...
Jianrong Li's user avatar
  • 6,101
0 votes
1 answer
236 views

Canonical basis of cluster algebras

Let $x_{k+1} = \frac{x_k^{d_k}+1}{x_{k-1}}$, $k \in \mathbb{Z}$, where $d_{k+2} = d_k \in \mathbb{Z_{>0}}$. Let $b=d_1$ and $c=d_2$. Define the cluster algebra $A = A(\left( \begin{matrix} 0 & ...
Jianrong Li's user avatar
  • 6,101
2 votes
1 answer
305 views

Cluster algebras of finite type

In the webpage, there is a result: Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...
Jianrong Li's user avatar
  • 6,101
4 votes
1 answer
203 views

Cluster algebra structure compatible with Poisson brackets

Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper. Suppose that we construct a maximal independent set of ...
Jianrong Li's user avatar
  • 6,101
2 votes
1 answer
213 views

Cluster algebra structure on the coordinate ring of $Mat_3$

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$. We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...
Jianrong Li's user avatar
  • 6,101
6 votes
1 answer
561 views

Proof of Laurent Phenomenon for Cluster Algebras

I went through the proof of the Laurent phenomenon for Cluster Algebras in Fomin and Zelevinsky's initial paper: Cluster Algebras I: Foundations. I am stuck at their claim that the gcd of two exchange ...
Confused's user avatar
3 votes
1 answer
262 views

Quiver folding and maximal green sequences

The technique of quiver folding (please see Folding by Automorphisms) can be used to prove statements about non-simply laced quivers (i.e. valued quivers) when they are already known in the simply-...
Ying Zhou's user avatar
  • 417
2 votes
0 answers
85 views

Characteristics of $c$-vectors of acyclic cluster algebras

In Speyer and Thomas's work, Acyclic Cluster Algebras Revisited the characteristics of $c$-vectors of cluster algebras with the $B$-matrix of the initial seed acyclic are given in Theorem 1.4. Do we ...
Ying Zhou's user avatar
  • 417
5 votes
2 answers
509 views

Do we have super Plucker relations for a super Grassmannian?

Super Grassmannians are introduced by Manin, see for example. We have Plucker relation for Grassmannian. Are there some references about super Plucker relations for super Grassmannian? Thank you ...
Jianrong Li's user avatar
  • 6,101
1 vote
1 answer
163 views

Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?

I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...
Ying Zhou's user avatar
  • 417
9 votes
1 answer
2k views

Cluster algebras and cluster varieties

I have a really basic question about cluster algebras and cluster varieties. According to the definition of Fomin-Zelevinsky a cluster algebra is generated by a bunch of polynomial rings inside the ...
Alexander Braverman's user avatar
1 vote
0 answers
95 views

Generalized Gaussian Decomposition

Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...
Daps's user avatar
  • 540
4 votes
3 answers
224 views

Cluster Variables for non-convex n-gons

Most of the lectures and lecture notes on Cluster Algebras (at least from Combinatorial point of view) start with mutations of the diagonals of a convex n-gon (mostly the pentagon) as the illustration ...
Kaveh's user avatar
  • 483
1 vote
1 answer
263 views

Number of cluster variables

In the paper Hernandez and Leclerc - Cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster ...
Jianrong Li's user avatar
  • 6,101
8 votes
0 answers
364 views

When is a $2$-Calabi–Yau triangulated category the cluster category of a QP?

Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the ...
Matthew Pressland's user avatar
4 votes
0 answers
214 views

``Occasional'' Laurent phenomenon

This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?). He asked ...
Alexey Ustinov's user avatar
5 votes
2 answers
645 views

How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$. Is there an algorithm which takes two ...
Mikhail's user avatar
  • 455
11 votes
1 answer
973 views

What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory? In my view, that means it should start off with unpunctured surfaces (and in fact,...
Hugh Thomas's user avatar
  • 6,075
8 votes
0 answers
221 views

Testing membership in a cluster algebra

Say I have a cluster algebra with principal coefficients and initial cluster $x_1,\ldots,x_n$. I don't want to invert the coefficient variables $y_1,\ldots,y_n$. The Laurent Phenomenon says that ...
Nathan Reading's user avatar
3 votes
2 answers
360 views

Kahler differentials on cluster varieties

On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
user36931's user avatar
  • 1,331
1 vote
1 answer
594 views

Finding particular reduced words for Weyl group elements

I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...
Matthew Pressland's user avatar
6 votes
0 answers
333 views

cluster variables and L-functions

There is something in common between cluster variables in the theory of cluster algebras, L-functions in number theory, namely the fact that both map direct sums to products, just like determinants ...
F. C.'s user avatar
  • 3,507
6 votes
0 answers
325 views

Non-crystallographic cluster algebras

Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
Philipp Lampe's user avatar
55 votes
1 answer
3k views

Intersecting family of triangulations

Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let $\...
Gil Kalai's user avatar
  • 24.2k
33 votes
2 answers
2k views

What do cluster algebras tell us about Grassmannians?

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
Matthew Pressland's user avatar
3 votes
0 answers
409 views

Motivic DT-Invariants for the Algebro-Geophobic

I am looking for as gentle of possible of an introduction to Kontsevich-Soibelman's theory of motivic DT-invariants. Specifically I am interested in the algebraic aspects of the theory and the ...
Steve's user avatar
  • 2,223
5 votes
0 answers
372 views

"Natural" Poisson structure on $(\mathbb{P}^1)^N$

Recently there is some interest in the Poisson geometry of "cluster manifolds", which are varieties associated to cluster algebras. See for example the works of Gekhtman, Shapiro and Vainshtein. In ...
Xin Nie's user avatar
  • 1,764
16 votes
0 answers
546 views

Catalan objects associated to a univariate polynomial

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data: a noncrossing matching on $2n$ ...
Hugh Thomas's user avatar
  • 6,075