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Questions tagged [cluster-algebra]

Use this tag for questions about constructively defined commutative rings equipped with a distinguished set of generators grouped into overlapping subsets of the same finite cardinality.

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Dimers from Postnikov diagrams

I am reading several papers on Postnikov diagrams, dimer models, and quivers. From the Postnikov diagram, we can draw a dimer model and an ice quiver. My question is as follows. Does every dimer model(...
aleph0's user avatar
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2 votes
0 answers
10 views

How to describe the partition of $GL_3(\mathbb{C})$ by Bruhat decomposition accurately?

We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic ...
YSouSerious's user avatar
1 vote
1 answer
66 views

A question about systems of nonlinear equations

Recently, I read a book on cluster algebra and come across a problem that could finally be reduced to a problem of solving a system of nonlinear equations. The question is: Give $b_{12},b_{13} , b_{...
fusheng's user avatar
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0 votes
1 answer
34 views

Why is this not a Postnikov diagram?

I am following this paper on Grassmannians and Cluster Structures. I drew the following diagram for $Gr(2,6)$: However, it doesn't satisfy the property that each alternating region is labelled by a $...
BulkyMolaMola's user avatar
1 vote
0 answers
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Fourier transform of unbounded self-adjoint operators

Given a nice function $f\colon\mathbb{R}\to\mathbb{S}^1\subset\mathbb{C}$ and an unbounded self-adjoint operator $P$ acting on the Hilbert space $L^2(\mathbb{R})$, one can define a unitary operator $f(...
Estwald's user avatar
  • 271
2 votes
1 answer
67 views

How to read off distinguished triangles and cluster-tilting objects in the cluster category of a Dynkin quiver?

I'm new to triangulated category and tilting theory. To illustrate, in $Q=A_4$ the module $M=kQ$ is cluster-tilting. While I know that $M$ satisfies $\mathrm{Ext}(M,M)=\mathrm{Hom}(M,M[1])=0$ by some ...
Richard Chen's user avatar
1 vote
0 answers
16 views

Question about the proof of the completeness of the category of rooted cluster algebras.

I have been reading the proof of Propostion 5.4 in On the category of rooted cluster algebras by Assem, Dupont and Schiffler in which the authors state that the category of rooted cluster algebra is ...
amator2357's user avatar
1 vote
1 answer
45 views

Using cluster categories to show that seed of acyclic cluster algebra is determined by the cluster

In corollary 2 of the article "From triangulated categories to cluster algebras II" by Caldero and Keller, it is stated that a seed $(u,Q)$ (where $u$ is a cluster and $Q$ a quiver) of an ...
VanD1206's user avatar
  • 429
1 vote
1 answer
177 views

Cluster algebras from quivers vs. cluster algebras from skew-symmetrizable matrices

In a course on cluster algebras we first defined the cluster algebra arising from a quiver. Next we saw that each quiver gives rise to an exchange matrix that is skew-symmetric. So we generalized to ...
Math's user avatar
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1 vote
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Toric charts from clusters of the coordinate ring of the Grassmannian

In his Paper [Sc], Scott proves that the coordinate ring of the grassmannian $ Gr(k,n) \subset \mathbb{P}\Lambda^k\mathbb{C}^n$, or rather of its affine cone, is a cluster algebra of geometric type. ...
Backfisch's user avatar
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Questions about $Y$-systems

Let $\Delta$ be a Dynkin diagram with vertex set $I$ and let $C$ be the Cartan matrix of $\Delta$ and $J$ be the identity matrix of the same size as $C$. The set $I$ of vertices is the disjoint union ...
billy192's user avatar
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2 votes
1 answer
235 views

Introduction to cluster algebras

I am doing an undergraduate research project and the topic I have been assigned is cluster algebras (an introduction). I was reviewing some bibliography and videos on the internet and the truth is I ...
Juan Daniel Valdivia Fuentes's user avatar
2 votes
0 answers
37 views

Ring automorphisms between cluster algebras of finite type $A$

To construct a cluster algebra (of finite type $A$) associated to a convex $m$-gon $P_m$ we first take a triangulation $T$ of $P_m$ and regard it as our initial seed with exchangeable variables being ...
billy192's user avatar
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Cluster as a transcendence basis of the field of rational functions.

Let $P$ be a free multiplicative abelian group of a finite rank $m$ with generators $g_1,\dots,g_m$. Field $F$ of rational functions in $n$ independent variables is set to be the ambient field in our ...
amator2357's user avatar
3 votes
1 answer
958 views

Example of a cluster variety

This question is basically just me asking for something to be either verified or rebutted. So I'm trying to work with cluster varieties, and no matter how much I look around, I simply am not ...
StormyTeacup's user avatar
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1 vote
0 answers
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Definition of cluster algebras.

I've been doing some reading on cluster algebra from surfaces and in Schiffler I encountered this Definition: Let $\mathcal{X}$ be the set of all cluster variables obtained by mutation from $(\...
amator2357's user avatar
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0 answers
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Cluster algebra associated to a d-gon

Recently I have been doing some reading on cluster algebras, for example this. When defining the cluster algebras associated to a $d$-gon, they claim that the cluster and coefficient variables of $A_{...
amator2357's user avatar
1 vote
1 answer
325 views

How exactly do I compute Poisson-Lie brackets?

This question comes from Example 4.2 of the Gekhtman-Shapiro-Vainshtein book Cluster Algebras and Poisson Geometry which I have attached. My goal is to understand how to compute $\{x_1’,x_2’\}=-\...
Jimmy Mixco's user avatar
3 votes
1 answer
123 views

What are the generating elements of a cluster algebra?

I'm trying to get into cluster algebras, and so have been looking at the basic stuff to get the foundations settled. A problem I've run into is that I've come across two (seemingly contradictory) ...
StormyTeacup's user avatar
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5 votes
2 answers
718 views

What are the good reading books to learn cluster algebra?

What are the good reading books to learn cluster algebra? I need a basic introductory books or notes in particular. I do not have any physics background and I want a book which starts with graph ...
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1 vote
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Why $U^w=U \cap B_- w B_-$ is parametrized by $\mathbb R^m$?

Let $G$ be a semisimple Lie group and $B_-$ a Borel subgroup of $G$. Let $U$ be a unipotent subgroup of $G$. In the paper, Proposition 1.1, it is said that $U^w=U \cap B_- w B_-$ is parametrized by $...
LJR's user avatar
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0 votes
0 answers
76 views

totally positive Grassmannians

On page 4 of this article https://arxiv.org/pdf/1608.05735.pdf is defined totally positive grassmanian, then he says that if an element $[z] \in {Gr_ {k, m}}$ is defined by a matrix $z$ full range $k \...
tomás's user avatar
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1 vote
0 answers
148 views

projective basis of a $P^2$ projective space

I'm working on Grassmanians, specifically on the Plucker embeding $p:G_{d,V}\rightarrow P(\wedge^2(V))$ where $V$ is a $k$-vector space 3-dimensional. Say {v_1,v_2,v_3} is a basis for $V$. Then we can ...
Irene's user avatar
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2 votes
0 answers
62 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
  • 1,160
30 votes
1 answer
2k 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}...
user avatar
1 vote
1 answer
217 views

Cluster algebra of finite type

It is proved in the paper that a cluster algebra is of finite type if its Cartan counter part of the principal part of its seeds is a Cartan matrix of finite type. If the initial quiver of a cluster ...
LJR's user avatar
  • 14.6k
0 votes
0 answers
126 views

Reference request: bounded derived categories and their Auslander-Reiten quivers

I have some knowledge of Auslander-Reiten theory, tilting theory, derived categories and triangulated categories though I still find most proofs using derived categories in "Tilting Theory and Cluster ...
Ying Zhou's user avatar
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0 answers
71 views

Mutations of an $A_n$ quiver to reach $A_n$ straight orientation

(1)For a quiver $A_n$ of arbitrary orientation does there exist a finite sequence of quiver mutations that can mutate it to $A_n$ straight orientation $1\rightarrow 2\rightarrow\cdots\rightarrow n$ or ...
Ying Zhou's user avatar
  • 760
1 vote
2 answers
105 views

How to understand that minors are matrix elements in fundamental representations of $SL_n$?

In the video, Lecture 3 of June 14, 49:00-53:00, it is said that "minors are matrix elements in fundamental representations of $SL_n$". What are fundamental representations of $SL_n$? How to ...
LJR's user avatar
  • 14.6k
4 votes
1 answer
188 views

Question about the definition of cluster algebras.

I have a question about the definition of a seed of a cluster algebra. It is said that a seed is a pair $(R, u)$, where $R$ is a quiver with $n$ vertices, $u = \{u_1, \ldots, u_n\}$ is a free ...
LJR's user avatar
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